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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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## 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

## 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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