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

## 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.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Balbharati textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

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.

Using Balbharati 12th Board Exam solutions Indefinite Integration exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Balbharati Solutions are important questions that can be asked in the final exam. Maximum students of Maharashtra State Board 12th Board Exam prefer Balbharati Textbook Solutions to score more in exam.

Get the free view of chapter 3 Indefinite Integration 12th Board Exam extra questions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board and can use Shaalaa.com to keep it handy for your exam preparation