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Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 3 - Indefinite Integration [Latest edition]

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Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board - Shaalaa.com
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Chapter 3: Indefinite Integration

Exercise 3.1Exercise 3.2 (A)Exercise 3.2 (B)Exercise 3.2 (C)Exercise 3.3Exercise 3.4Miscellaneous Exercise 3
Exercise 3.1 [Page 102]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Indefinite Integration Exercise 3.1 [Page 102]

Exercise 3.1 | Q 1.1 | Page 102

Integrate the following w.r.t. x : x3 + x2 – x + 1

Exercise 3.1 | Q 1.2 | Page 102

Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`

Exercise 3.1 | Q 1.3 | Page 102

Integrate the following w.r.t. x : `3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`

Exercise 3.1 | Q 1.4 | Page 102

Integrate the following w.r.t. x : `2x^3 - 5x + 3/x + 4/x^5`

Exercise 3.1 | Q 1.5 | Page 102

Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`

Exercise 3.1 | Q 2.01 | Page 102

Evaluate the following integrals : tan2x

Exercise 3.1 | Q 2.02 | Page 102

Evaluate the following integrals : `int (sin2x)/(cosx)dx`

Exercise 3.1 | Q 2.03 | Page 102

Evaluate the following integrals : `int sin x/cos^2x dx`

Exercise 3.1 | Q 2.04 | Page 102

Evaluate the following integrals : `int (cos2x)/sin^2x dx` 

Exercise 3.1 | Q 2.05 | Page 102

Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`

Exercise 3.1 | Q 2.06 | Page 102

Evaluate the following integrals : `int sinx/(1 + sinx)dx`

Exercise 3.1 | Q 2.07 | Page 102

Evaluate the following integrals : `int tanx/(sec x + tan x)dx`

Exercise 3.1 | Q 2.08 | Page 102

Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`

Exercise 3.1 | Q 2.09 | Page 102

Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`

Exercise 3.1 | Q 2.1 | Page 102

Evaluate the following integrals : `int sin 4x cos 3x dx`

Exercise 3.1 | Q 3.01 | Page 102

Evaluate the following integrals : `int x/(x + 2).dx`

Exercise 3.1 | Q 3.02 | Page 102

Evaluate the following integrals : `int(4x + 3)/(2x + 1).dx`

Exercise 3.1 | Q 3.03 | Page 102

Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`

Exercise 3.1 | Q 3.04 | Page 102

Evaluate the following integrals : `int(x - 2)/sqrt(x + 5).dx`

Exercise 3.1 | Q 3.05 | Page 102

Evaluate the following integrals : `int (2x - 7)/sqrt(4x - 1).dx`

Exercise 3.1 | Q 3.06 | Page 102

Evaluate the following integrals : `int (sin4x)/(cos2x).dx`

Exercise 3.1 | Q 3.07 | Page 102

Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`

Exercise 3.1 | Q 3.08 | Page 102

Evaluate the following integrals : `int cos^2x.dx`

Exercise 3.1 | Q 3.09 | Page 102

Evaluate the following integrals : `int(2)/(sqrt(x) - sqrt(x + 3)).dx`

Exercise 3.1 | Q 3.1 | Page 102

Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`

Exercise 3.1 | Q 4 | Page 102

If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)

Exercise 3.2 (A) [Page 110]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Indefinite Integration Exercise 3.2 (A) [Page 110]

Exercise 3.2 (A) | Q 1.01 | Page 110

Integrate the following functions w.r.t. x : `(logx)^n/x`

Exercise 3.2 (A) | Q 1.02 | Page 110

Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`

Exercise 3.2 (A) | Q 1.03 | Page 110

Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`

Exercise 3.2 (A) | Q 1.04 | Page 110

Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`

Exercise 3.2 (A) | Q 1.05 | Page 110

Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`

Exercise 3.2 (A) | Q 1.06 | Page 110

Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`

Exercise 3.2 (A) | Q 1.07 | Page 110

Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`

Exercise 3.2 (A) | Q 1.08 | Page 110

Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`

Exercise 3.2 (A) | Q 1.09 | Page 110

Integrate the following functions w.r.t. x : sin4x.cos3x

Exercise 3.2 (A) | Q 1.1 | Page 110

Integrate the following functions w.r.t. x : `(1)/(4x + 5x^-11)`

Exercise 3.2 (A) | Q 1.11 | Page 110

Integrate the following functions w.r.t. x : x9.sec2(x10)

Exercise 3.2 (A) | Q 1.12 | Page 110

Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1 

Exercise 3.2 (A) | Q 1.13 | Page 110

Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`

Exercise 3.2 (A) | Q 1.14 | Page 110

Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`

Exercise 3.2 (A) | Q 1.15 | Page 110

Integrate the following functions w.r.t. x : `(2sin x cos x)/(3 cos^2x + 4 sin^2 x)`

Exercise 3.2 (A) | Q 1.16 | Page 110

Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`

Exercise 3.2 (A) | Q 1.17 | Page 110

Integrate the following functions w.r.t. x : `(10x^9  10^x.log10)/(10^x + 10^10)`

Exercise 3.2 (A) | Q 1.18 | Page 110

Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`

Exercise 3.2 (A) | Q 1.19 | Page 110

Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`

Exercise 3.2 (A) | Q 1.2 | Page 110

Integrate the following functions w.r.t. x : `x^5sqrt(a^2 + x^2)`

Exercise 3.2 (A) | Q 1.21 | Page 110

Integrate the following functions w.r.t. x : `(5 - 3x)(2 - 3x)^(-1/2)`

Exercise 3.2 (A) | Q 1.22 | Page 110

Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`

Exercise 3.2 (A) | Q 1.23 | Page 110

Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`

Exercise 3.2 (A) | Q 1.24 | Page 110

Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`

Exercise 3.2 (A) | Q 1.25 | Page 110

Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.

Exercise 3.2 (A) | Q 2.01 | Page 110

Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`

Exercise 3.2 (A) | Q 2.02 | Page 110

Integrate the following functions w.r.t. x : `cosx/sin(x - a)`

Exercise 3.2 (A) | Q 2.03 | Page 110

Integrate the following functions w.r.t. x : `sin(x - a)/cos(x  + b)`

Exercise 3.2 (A) | Q 2.04 | Page 110

Integrate the following functions w.r.t. x : `(1)/(sinx.cosx + 2cos^2x)`

Exercise 3.2 (A) | Q 2.05 | Page 110

Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`

Exercise 3.2 (A) | Q 2.06 | Page 110

Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`

Exercise 3.2 (A) | Q 2.07 | Page 110

Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`

Exercise 3.2 (A) | Q 2.08 | Page 110

Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`

Exercise 3.2 (A) | Q 2.09 | Page 110

Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`

Exercise 3.2 (A) | Q 2.1 | Page 110

Integrate the following functions w.r.t. x : cos8xcotx

Exercise 3.2 (A) | Q 2.11 | Page 110

Integrate the following functions w.r.t. x : tan5x

Exercise 3.2 (A) | Q 2.12 | Page 110

Integrate the following functions w.r.t. x : cos7x

Exercise 3.2 (A) | Q 2.13 | Page 110

Integrate the following functions w.r.t. x :  tan 3x tan 2x tan x

Exercise 3.2 (A) | Q 2.14 | Page 110

Integrate the following functions w.r.t. x : sin5x.cos8x

Exercise 3.2 (A) | Q 2.15 | Page 110

Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`

Exercise 3.2 (A) | Q 2.16 | Page 110

Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`

Exercise 3.2 (A) | Q 2.17 | Page 110

Integrate the following functions w.r.t. x : `(sinx cos^3x)/(1 + cos^2x)`

Exercise 3.2 (B) [Page 123]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Indefinite Integration Exercise 3.2 (B) [Page 123]

Exercise 3.2 (B) | Q 1.01 | Page 123

Evaluate the following : `int (1)/(4x^2 - 3).dx`

Exercise 3.2 (B) | Q 1.02 | Page 123

Evaluate the following : `int (1)/(25 - 9x^2).dx`

Exercise 3.2 (B) | Q 1.03 | Page 123

Evaluate the following : `int (1)/(7 + 2x^2).dx`

Exercise 3.2 (B) | Q 1.04 | Page 123

Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`

Exercise 3.2 (B) | Q 1.05 | Page 123

Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`

Exercise 3.2 (B) | Q 1.06 | Page 123

Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`

Exercise 3.2 (B) | Q 1.07 | Page 123

Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`

Exercise 3.2 (B) | Q 1.08 | Page 123

Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`

Exercise 3.2 (B) | Q 1.09 | Page 123

Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`

Exercise 3.2 (B) | Q 1.1 | Page 123

Evaluate the following : `int  (1)/(x^2 + 8x + 12).dx`

Exercise 3.2 (B) | Q 1.11 | Page 123

Evaluate the following : `int (1)/(1 + x - x^2).dx`

Exercise 3.2 (B) | Q 1.12 | Page 123

Evaluate the following : `(1)/(4x^2 - 20x + 17)`

Exercise 3.2 (B) | Q 1.13 | Page 123

Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`

Exercise 3.2 (B) | Q 1.14 | Page 123

Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`

Exercise 3.2 (B) | Q 1.15 | Page 123

Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`

Exercise 3.2 (B) | Q 1.16 | Page 123

Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`

Exercise 3.2 (B) | Q 1.17 | Page 123

Evaluate the following : `int (1)/sqrt((x - 3)(x + 2)).dx`

Exercise 3.2 (B) | Q 1.18 | Page 123

Evaluate the following : `int (1)/(4 + 3cos^2x).dx`

Exercise 3.2 (B) | Q 1.19 | Page 123

Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`

Exercise 3.2 (B) | Q 1.2 | Page 123

Evaluate the following : `int sinx/(sin 3x).dx`

Exercise 3.2 (B) | Q 2.1 | Page 123

Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`

Exercise 3.2 (B) | Q 2.2 | Page 123

Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`

Exercise 3.2 (B) | Q 2.3 | Page 123

Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`

Exercise 3.2 (B) | Q 2.4 | Page 123

Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`

Exercise 3.2 (B) | Q 2.5 | Page 123

Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`

Exercise 3.2 (B) | Q 2.6 | Page 123

Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`

Exercise 3.2 (B) | Q 2.7 | Page 123

Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`

Exercise 3.2 (B) | Q 2.8 | Page 123

Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`

Exercise 3.2 (B) | Q 2.9 | Page 123

Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`

Exercise 3.2 (C) [Page 128]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Indefinite Integration Exercise 3.2 (C) [Page 128]

Exercise 3.2 (C) | Q 1.1 | Page 128

Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`

Exercise 3.2 (C) | Q 1.2 | Page 128

Evaluate the following integrals : `int (2x + 1)/(x^2 + 4x - 5).dx`

Exercise 3.2 (C) | Q 1.3 | Page 128

Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`

Exercise 3.2 (C) | Q 1.4 | Page 128

Evaluate the following integrals :  `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`

Exercise 3.2 (C) | Q 1.5 | Page 128

x + Evaluate the following integrals : `int (7x + 3)/sqrt(3 + 2x - x^2).dx`

Exercise 3.2 (C) | Q 1.6 | Page 128

Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`

Exercise 3.2 (C) | Q 1.7 | Page 128

Evaluate the following integrals : `int sqrt((9 - x)/x).dx`

Exercise 3.2 (C) | Q 1.8 | Page 128

Evaluate the following integrals : `int (3cosx)/(4sin^2x + 4sinx - 1).dx`

Exercise 3.2 (C) | Q 1.9 | Page 128

Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`

Exercise 3.3 [Pages 137 - 138]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Indefinite Integration Exercise 3.3 [Pages 137 - 138]

Exercise 3.3 | Q 1.01 | Page 137

Evaluate the following : `int x^2.log x.dx`

Exercise 3.3 | Q 1.02 | Page 137

Evaluate the following : `int x^2 sin 3x  dx`

Exercise 3.3 | Q 1.03 | Page 137

Evaluate the following : `int x tan^-1 x .dx`

Exercise 3.3 | Q 1.04 | Page 137

Evaluate the following : `int x^2tan^-1x.dx`

Exercise 3.3 | Q 1.05 | Page 137

Evaluate the following : `int x^3.tan^-1x.dx`

Exercise 3.3 | Q 1.06 | Page 137

Evaluate the following : `int (logx)2.dx`

Exercise 3.3 | Q 1.07 | Page 137

Evaluate the following : `int sec^3x.dx`

Exercise 3.3 | Q 1.08 | Page 137

Evaluate the following : `int x.sin^2x.dx`

Exercise 3.3 | Q 1.09 | Page 137

Evaluate the following : `int x^3.logx.dx`

Exercise 3.3 | Q 1.1 | Page 137

Evaluate the following : `int e^(2x).cos 3x.dx`

Exercise 3.3 | Q 1.11 | Page 137

Evaluate the following : `int x.sin^2 x.dx`

Exercise 3.3 | Q 1.12 | Page 137

Evaluate the following : `int x^2*cos^-1 x*dx`

Exercise 3.3 | Q 1.13 | Page 137

Evaluate the following : `int log(logx)/x.dx`

Exercise 3.3 | Q 1.14 | Page 137

Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`

Exercise 3.3 | Q 1.15 | Page 137

Evaluate the following : `int cos sqrt(x).dx`

Exercise 3.3 | Q 1.16 | Page 137

Evaluate the following : `int sin θ.log (cos θ).dθ`

Exercise 3.3 | Q 1.17 | Page 137

Evaluate the following : `int x.cos^3x.dx`

Exercise 3.3 | Q 1.18 | Page 137

Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`

Exercise 3.3 | Q 1.19 | Page 137

Evaluate the following : `int logx/x.dx`

Exercise 3.3 | Q 1.2 | Page 137

Evaluate the following : `int x.sin 2x. cos 5x.dx`

Exercise 3.3 | Q 1.21 | Page 137

Evaluate the following : `int cos(root(3)(x)).dx`

Exercise 3.3 | Q 2.01 | Page 138

Integrate the following functions w.r.t. x : `e^(2x).sin3x`

Exercise 3.3 | Q 2.02 | Page 138

Integrate the following functions w.r.t. x : `e^-x cos2x`

Exercise 3.3 | Q 2.03 | Page 138

Integrate the following functions w.r.t. x : sin (log x)

Exercise 3.3 | Q 2.04 | Page 138

Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`

Exercise 3.3 | Q 2.05 | Page 138

Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`

Exercise 3.3 | Q 2.06 | Page 138

Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`

Exercise 3.3 | Q 2.07 | Page 138

Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`

Exercise 3.3 | Q 2.08 | Page 138

Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`

Exercise 3.3 | Q 2.09 | Page 138

Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`

Exercise 3.3 | Q 2.1 | Page 138

Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`

Exercise 3.3 | Q 2.11 | Page 138

Integrate the following functions w.r.t. x : `sqrt(x^2 + 2x + 5)`

Exercise 3.3 | Q 2.12 | Page 138

Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`

Exercise 3.3 | Q 3.1 | Page 138

Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]e 

Exercise 3.3 | Q 3.2 | Page 138

Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`

Exercise 3.3 | Q 3.3 | Page 138

Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`

Exercise 3.3 | Q 3.4 | Page 138

Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`

Exercise 3.3 | Q 3.5 | Page 138

Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`

Exercise 3.3 | Q 3.6 | Page 138

Integrate the following functions w.r.t. x : `e^(5x).[(5x.logx + 1)/x]`

Exercise 3.3 | Q 3.7 | Page 138

Integrate the following functions w.r.t. x : `e^(sin^-1x).[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`

Exercise 3.3 | Q 3.8 | Page 138

Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`

Exercise 3.3 | Q 3.9 | Page 138

Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)] 

Exercise 3.4 [Pages 144 - 145]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Indefinite Integration Exercise 3.4 [Pages 144 - 145]

Exercise 3.4 | Q 1.01 | Page 144

Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`

Exercise 3.4 | Q 1.02 | Page 144

Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`

Exercise 3.4 | Q 1.03 | Page 144

Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`

Exercise 3.4 | Q 1.04 | Page 145

Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`

Exercise 3.4 | Q 1.05 | Page 145

Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`

Exercise 3.4 | Q 1.06 | Page 145

Integrate the following w.r.t. x : `(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`

Exercise 3.4 | Q 1.07 | Page 145

Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`

Exercise 3.4 | Q 1.08 | Page 145

Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`

Exercise 3.4 | Q 1.09 | Page 145

Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`

Exercise 3.4 | Q 1.1 | Page 145

Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`

Exercise 3.4 | Q 1.11 | Page 145

Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`

Exercise 3.4 | Q 1.12 | Page 145

Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`

Exercise 3.4 | Q 1.13 | Page 145

Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`

Exercise 3.4 | Q 1.14 | Page 145

Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`

Exercise 3.4 | Q 1.15 | Page 145

Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`

Exercise 3.4 | Q 1.16 | Page 145

Integrate the following w.r.t. x : `(1)/(x^3 - 1)`

Exercise 3.4 | Q 1.17 | Page 145

Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`

Exercise 3.4 | Q 1.18 | Page 145

Integrate the following w.r.t. x : `(1)/(sinx + sin2x)`

Exercise 3.4 | Q 1.19 | Page 145

Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`

Exercise 3.4 | Q 1.2 | Page 145

Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`

Exercise 3.4 | Q 1.21 | Page 145

Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`

Exercise 3.4 | Q 1.22

Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`

Exercise 3.4 | Q 1.23 | Page 145

Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`

Miscellaneous Exercise 3 [Pages 148 - 150]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Indefinite Integration Miscellaneous Exercise 3 [Pages 148 - 150]

Miscellaneous Exercise 3 | Q 1.01 | Page 148

Choose the correct option from the given alternatives : 

`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =

  • `(1)/(2)sqrt(x + 1) + c`

  • `(2)/(3)(x + 1)^(3/2) + c`

  • `sqrt(x + 1) + c`

  • `2(x - 1)^(3/2) + c`

Miscellaneous Exercise 3 | Q 1.02 | Page 148

Choose the correct options from the given alternatives :

`int (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =

  • log x – f(x) + c

  • f(x) + log x + c

  • f(x) – log  x + c

  • `(1)/(5) x^5f(x) + c`

Miscellaneous Exercise 3 | Q 1.03 | Page 148

Choose the correct options from the given alternatives :

`int (log (3x))/(xlog (9x))*dx` =

  • log (3x) – log (9x) + c·

  • log (x) – (log 3) · log (log 9x) + c

  • log 9 – (log x) · log (log 3x) + c

  • log (x) + (log 3) · log (log 9x) + c

Miscellaneous Exercise 3 | Q 1.04 | Page 148

Choose the correct options from the given alternatives :

`int (sin^m x)/(cos^(m+2)x)*dx` = 

  • `(tan^(m+1)x)/(m + 1) + c`

  • (m + 2)tanm+1 x + c

  • `tan^mx/m + c`

  • (m + 1)tanm+1 x + c

Miscellaneous Exercise 3 | Q 1.05 | Page 148

Choose the correct options from the given alternatives :

`int tan(sin^-1 x)*dx` =

  • `(1 - x^2)^(-1/2) + c`

  • `(1 - x^2)^(1/2) + c`

  • `tan^m x/sqrt(1 - x^2) + c`

  • `- sqrt(1 - x^2) + c`

Miscellaneous Exercise 3 | Q 1.06 | Page 148

Choose the correct options from the given alternatives :

`int (x- sinx)/(1 - cosx)*dx` =

  • `x cot (x/2) + c`

  • `- x cot (x/2) + c`

  • `cot (x/2) + c`

  • `x tan (x/2) + c`

Miscellaneous Exercise 3 | Q 1.07 | Page 148

Choose the correct options from the given alternatives :

If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` =

  • `e^(sin^-1x)*(sin^-1 x - 1) + c`

  • `e^(sin^-1x)*(1 - sin^-1x) + c`

  • `e^(sin^-1x)*(sin^-1 x + 1) + c`

  • `-e^(sin^-1x)*(sin^-1 x + 1) + c`

Miscellaneous Exercise 3 | Q 1.08 | Page 148

Choose the correct options from the given alternatives :

If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =

  • (5, 3)

  • (3, 5)

  • `(1/5, 1/3)`

  • (4, 4)

Miscellaneous Exercise 3 | Q 1.09 | Page 149

Choose the correct options from the given alternatives :

`int (1)/(cosx - cos^2x)*dx` =

  • `log ("cosec"x - cotx) + tan(x/2) + c`

  • sin 2x – cos x + c

  • `log (secx + tanx) - cot(x/2) + c`

  • cos 2x – sin x + c

Miscellaneous Exercise 3 | Q 1.1 | Page 149

Choose the correct options from the given alternatives :

`int sqrt(cotx)/(sinx*cosx)*dx` =

  • `2sqrt(cotx) + c`

  • `-2sqrt(cotx) + c`

  • `(1)/(2)sqrt(cotx) + c`

  • `sqrt(cotx) + c`

Miscellaneous Exercise 3 | Q 1.11 | Page 149

Choose the correct options from the given alternatives :

`int (e^x(x - 1))/x^2*dx` =

  • `e^x/x + c`

  • `e^x/x^2 + c`

  • `(x - 1/x)e^x + c`

  • `xe^-x + c`

Miscellaneous Exercise 3 | Q 1.12 | Page 149

Choose the correct options from the given alternatives :

`int sin (log x)*dx` =

  • `x/(2)[sin (log x) - cos (log x)] + c`

  • `x/(2)[sin (log x) + cos (log x)] + c`

  • `x/(2)[cos (log x) - sin (log x)] + c`

  • `x/(4)[cos (log x) - sin (log x)] + c`

Miscellaneous Exercise 3 | Q 1.13 | Page 149

Choose the correct options from the given alternatives :

`int f x^x (1 + log x)*dx`

  • `(1)/(2)(1 + log x)^2 + c`

  • x2x + c

  • xx log x + c

  • xx + c

Miscellaneous Exercise 3 | Q 1.14 | Page 149

Choose the correct options from the given alternatives :

`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =

  • `log (sin^(-4/7) x) + c`

  • `(4)/(7)tan^(4/7) x + c`

  • `-(7)/(4)tan^(-4/7) x + c`

  • `log (cos^(3/7) x) + c`

Miscellaneous Exercise 3 | Q 1.15 | Page 149

Choose the correct options from the given alternatives :

`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =

  • sin 2x + c

  • cos 2x + c

  • tan 2x + c

  • 2 sin 2x + c

Miscellaneous Exercise 3 | Q 1.16 | Page 149

Choose the correct options from the given alternatives : 

`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =

  • `log(tanx - sqrt(tan^2x - 1)) + c`

  • sin–1 (tan x) + c

  • 1 + sin–1 (cot x) + c

  • `log(tanx + sqrt(tan^2x - 1)) + c`

Miscellaneous Exercise 3 | Q 1.17 | Page 150

Choose the correct options from the given alternatives :

`int logx/(log ex)^2*dx` =

  • `x/(1 + log x) + c`

  • x(1 + log x) + c

  • `x/(1 + log x) + c`

  • `x/(1 - log x) + c`

Miscellaneous Exercise 3 | Q 1.18 | Page 150

Choose the correct options from the given alternatives :

`int [sin (log x) + cos (log x)]*dx` =

  • x cos (log x) + c

  • sin (log x) + c

  • cos (log x) + c

  • x sin (log x) + c

Miscellaneous Exercise 3 | Q 1.19 | Page 150

Choose the correct options from the given alternatives :

`int (cos2x - 1)/(cos2x + 1)*dx` =

  • tan x – x + c

  • x + tan x + c

  • x – tan x + c

  • – x – cot x + c

Miscellaneous Exercise 3 | Q 1.2 | Page 150

Choose the correct options from the given alternatives :

`int (e^(2x) + e^-2x)/e^x*dx` =

  • `e^x - (1)/(3e^(3x)) + c`

  • `e^x + (1)/(3e^(3x)) + c`

  • `e^-x + (1)/(3e^(3x)) + c`

  • `e^-x - (1)/(3e^(3x)) + c`

Miscellaneous Exercise 3 | Q 2.1 | Page 150

Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`

Miscellaneous Exercise 3 | Q 2.2 | Page 150

integrate the following with respect to the respective variable : `x^2/(x + 1)`

Miscellaneous Exercise 3 | Q 2.3 | Page 150

Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`

Miscellaneous Exercise 3 | Q 2.4 | Page 150

Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`

Miscellaneous Exercise 3 | Q 2.5 | Page 150

Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`

Miscellaneous Exercise 3 | Q 2.6 | Page 150

Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`

Miscellaneous Exercise 3 | Q 2.7 | Page 150

Integrate the following with respect to the respective variable : cos 3x cos 2x cos x

Miscellaneous Exercise 3 | Q 2.8 | Page 150

Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`

Miscellaneous Exercise 3 | Q 2.9 | Page 150

Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`

Miscellaneous Exercise 3 | Q 3.01 | Page 150

Integrate the following w.r.t. x: `(1 + log x)^2/x`

Miscellaneous Exercise 3 | Q 3.02 | Page 150

Integrate the following w.r.t.x : cot–1 (1 – x + x2)

Miscellaneous Exercise 3 | Q 3.03 | Page 150

Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`

Miscellaneous Exercise 3 | Q 3.04 | Page 150

Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`

Miscellaneous Exercise 3 | Q 3.05 | Page 150

Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`

Miscellaneous Exercise 3 | Q 3.06 | Page 150

Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`

Miscellaneous Exercise 3 | Q 3.07 | Page 150

Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`

Miscellaneous Exercise 3 | Q 3.08 | Page 150

Integrate the following w.r.t.x : log (log x)+(log x)–2 

Miscellaneous Exercise 3 | Q 3.09 | Page 150

Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`

Miscellaneous Exercise 3 | Q 3.1 | Page 150

Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`

Miscellaneous Exercise 3 | Q 3.11 | Page 150

Integrate the following w.r.t.x : `(3x + 1)/sqrt(-2x^2 + x + 3)`

Miscellaneous Exercise 3 | Q 3.12 | Page 150

Integrate the following w.r.t.x : log (x2 + 1)

Miscellaneous Exercise 3 | Q 3.13 | Page 150

Integrate the following w.r.t.x : e2x sin x cos x

Miscellaneous Exercise 3 | Q 3.14 | Page 150

Integrate the following w.r.t.x : `x^2/((x - 1)(3x - 1)(3x - 2)`

Miscellaneous Exercise 3 | Q 3.15 | Page 150

Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`

Miscellaneous Exercise 3 | Q 3.16 | Page 150

Integrate the following w.r.t.x :  `sec^2x sqrt(7 + 2 tan x - tan^2 x)`

Miscellaneous Exercise 3 | Q 3.17 | Page 150

Integrate the following w.r.t.x : `(x + 5)/(x^3 + 3x^2 - x - 3)`

Miscellaneous Exercise 3 | Q 3.18 | Page 150

Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`

Miscellaneous Exercise 3 | Q 3.19 | Page 150

Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`

Miscellaneous Exercise 3 | Q 3.2 | Page 150

Integrate the following w.r.t.x : sec4x cosec2x

Chapter 3: Indefinite Integration

Exercise 3.1Exercise 3.2 (A)Exercise 3.2 (B)Exercise 3.2 (C)Exercise 3.3Exercise 3.4Miscellaneous Exercise 3
Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board - Shaalaa.com

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 3 - Indefinite Integration

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 3 (Indefinite Integration) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 3 Indefinite Integration are Indefinite Integration, Methods of Integration: Integration by Substitution, Methods of Integration: Integration by Parts, Methods of Integration: Integration Using Partial Fractions.

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