Contents

- 1 NCERT Solutions for Class 12 Maths Chapter 7 Integrals
- 1.1 Chapter 7 Integrals Exercise 7.1
- 1.2 Chapter 7 Integrals Exercise 7.2
- 1.3 Chapter 7 Integrals Exercise 7.3
- 1.4 Chapter 7 Integrals Exercise 7.4
- 1.5 Chapter 7 Integrals Exercise 7.5
- 1.6 Chapter 7 Integrals Exercise 7.6
- 1.7 Chapter 7 Integrals Exercise 7.7
- 1.8 Chapter 7 Integrals Exercise 7.8
- 1.9 Chapter 7 Integrals Exercise 7.9
- 1.10 Chapter 7 Integrals Exercise 7.10
- 1.11 Chapter 7 Integrals Exercise 7.11

These Solutions are part of NCERT Solutions for Class 12 Maths . Here we have given NCERT Solutions for Class 12 Maths Chapter 7 Integrals

Board |
CBSE |

Textbook |
NCERT |

Class |
Class 12 |

Subject |
Maths |

Chapter |
Chapter 7 |

Chapter Name |
Integrals |

Exercise |
Ex 7.1, Ex 7.2, Ex 7.3, Ex 7.4, Ex 7.5,Ex 7.6, Ex 7.7, Ex 7.8, Ex 7.9, Ex 7.10, Ex 7.11 |

Number of Questions Solved |
228 |

Category |
NCERT Solutions |

## NCERT Solutions for Class 12 Maths Chapter 7 Integrals

### Chapter 7 Integrals Exercise 7.1

**Find an antiderivative (or integral) of the following by the method of inspection:**

**Question 1.**

sin 2x

**Solution:**

**Question 2.**

cos 3x

**Solution:**

**Question 3.**

**Solution:**

**Question 4.**

(ax + c)²

**Solution:**

**Question 5.**

**Solution:**

**Find the following integrals in Exercises 6 to 20 :**

**Question 6.**

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

**Solution:**

**Question 10.**

**Solution:**

**Question 11.**

**Solution:**

**Question 12.**

**Solution:**

**Question 13.**

**Solution:**

**Question 14.**

**Solution:**

**Question 15.**

**Solution:**

**Question 16.**

**Solution:**

**Question 17.**

**Solution:**

**Question 18.**

**Solution:**

= tanx + secx + c

**Question 19.**

**Solution:**

**Question 20.**

**Solution:**

= 2tanx – 3secx + c

**Choose the correct answer in Exercises 21 and 22.**

**Question 21.**

The antiderivative equals

(a)

(b)

(c)

(d)

**Solution:**

(c)

**Question 22.**

If such that f(2)=0 then f(x) is

(a)

(b)

(c)

(d)

**Solution:**

(a)

### Chapter 7 Integrals Exercise 7.2

**Integrate the functions in Exercises 1 to 37:**

**Question 1.**

**Solution:**

Let 1+x² = t

⇒ 2xdx = dt

**Question 2.**

**Solution:**

Let logx = t

⇒

**Question 3.**

**Solution:**

Put 1+logx = t

∴

= log|1+logx|+c

**Question 4.**

sinx sin(cosx)

**Solution:**

Put cosx = t, -sinx dx = dt

**Question 5.**

sin(ax+b) cos(ax+b)

**Solution:**

let sin(ax+b) = t

⇒ cos(ax+b)dx = dt

**Question 6.**

**Solution:**

**Question 7.**

**Solution:**

Let x+2 = t²

⇒ dx = 2t dt

**Question 8.**

**Solution:**

let 1+2x² = t²

⇒ 4x dx = 2t dt

**Question 9.**

**Solution:**

let x²+x+1 = t

⇒(2x+1)dx = dt

**Question 10.**

**Solution:**

Let √x-1 = t

= 2logt + c

= 2log(√x-1)+c

**Question 11.**

**Solution:**

let x+4 = t

⇒ dx = dt, x = t-4

**Question 12.**

**Solution:**

**Question 13.**

**Solution:**

Let 2+3x³ = t

⇒ 9x² dx = dt

**Question 14.**

**Solution:**

Put log x = t, so that

**Question 15.**

**Solution:**

put 9-4x² = t, so that -8x dx = dt

**Question 16.**

**Solution:**

put 2x+3 = t

so that 2dx = dt

**Question 17.**

**Solution:**

Let x² = t

⇒ 2xdx = dt ⇒

**Question 18.**

**Solution:**

**Question 19.**

**Solution:**

put e^{x}+e^{-x} = t

so that (e^{x}-e^{-x})dx = dt

**Question 20.**

**Solution:**

put e^{2x}-e^{-2x} = t

so that (2e^{2x}-2e^{-2x})dx = dt

**Question 21.**

tan²(2x-3)

**Solution:**

∫tan²(2x-3)dx = ∫[sec²(2x-3)-1]dx = I

put 2x-3 = t

so that 2dx = dt

I = ∫sec²t dt-x+c

=

=

**Question 22.**

sec²(7-4x)

**Solution:**

∫sec²(7-4x)dx

=

**Question 23.**

**Solution:**

**Question 24.**

**Solution:**

put 2sinx+4cosx = t

⇒ (2cosx-3sinx)dx = dt

**Question 25.**

**Solution:**

put 1-tanx = t

so that -sec²x dx = dt

**Question 26.**

**Solution:**

= 2sin√x+c

**Question 27.**

**Solution:**

put sin2x = t²

⇒ cos2x dx = t dt

**Question 28.**

**Solution:**

put 1+sinx = t²

⇒cosx dx = 2t dt

**Question 29.**

cotx log sinx

**Solution:**

put log sinx = t,

⇒ cot x dx = dt

**Question 30.**

**Solution:**

put 1+cosx = t

⇒ -sinx dx = dt

=-log(1+cosx)+c

**Question 31.**

**Solution:**

put 1+cosx = t

so that -sinx dx = dt

**Question 32.**

**Solution:**

**Question 33.**

**Solution:**

**Question 34.**

**Solution:**

**Question 35.**

**Solution:**

let 1+logx = t

⇒

**Question 36.**

**Solution:**

put x+logx = t

**Question 37.**

**Solution:**

**Choose the correct answer in exercises 38 and 39**

**Question 38.**

(a) 10^{x} – x^{10} + C

(b) 10^{x} + x^{10} + C

(c) (10^{x} – x^{10}) + C

(d) log (10^{x} + x^{10}) + C

**Solution:**

(d)

= log (10^{x} + x^{10}) + C

**Question 39.**

(a) tanx + cotx + c

(b) tanx – cotx + c

(c) tanx cotx + c

(d) tanx – cot2x + c

**Solution:**

(c)

= tanx – cotx + c

### Chapter 7 Integrals Exercise 7.3

**Find the integrals of the functions in Exercises 1 to 22.**

**Question 1.**

sin²(2x+5)

**Solution:**

∫sin²(2x+5)dx

= ∫[1-cos2(2x+5)]dx

= ∫[1-cos(4x+10)]dx

=

**Question 2.**

sin3x cos4x

**Solution:**

∫sin3x cos4x

= ∫[sin(3x+4x)+cos(3x-4x)]dx

= ∫[sin7x+sin(-x)]dx

=

**Question 3.**

∫cos2x cos4x cos6x dx

**Solution:**

∫cos2x cos4x cos6x dx

= ∫(cos6x+cos2x) cos6x dx

**Question 4.**

∫sin^{3}(2x+1)dx

**Solution:**

= ∫[3sin(2x+1)-sin3(2x+1)]dx

=

=

**Question 5.**

sin^{3}x cos^{3}x

**Solution:**

put sin x = t

⇒ cos x dx = dt

**Question 6.**

sinx sin2x sin3x

**Solution:**

∫sinx sin2x sin3x dx

= ∫ 2sin x sin 2x sin 3x dx

= ∫ (cosx – cos3x)sin 3x dx

= ∫ (sin 4x + sin 2x – sin 6x)dx

=

**Question 7.**

sin 4x sin 8x

**Solution:**

∫sin 4x sin 8xdx

= ∫(cos 4x – cos 12x)dx

=

**Question 8.**

**Solution:**

**Question 9.**

**Solution:**

**Question 10.**

∫sinx^{4} dx

**Solution:**

**Question 11.**

cos^{4} 2x

**Solution:**

∫ cos^{4} 2x dx

**Question 12.**

**Solution:**

**Question 13.**

**Solution:**

let I =

= 2∫cos x dx + 2cos α∫dx

= 2(sinx+xcosα)+c

**Question 14.**

**Solution:**

let I =

put cosx+sinx = t

⇒ (-sinx+cosx)dx = dt

**Question 15.**

**Solution:**

I = ∫(sec^{2}2x-1)sec2x tan 2xdx

put sec2x=t,2 sec2x tan2x dx=dt

**Question 16.**

tan^{4}x

**Solution:**

let I = ∫tan^{4} dx

= ∫(sec²x-1)²dx

**Question 17.**

**Solution:**

= secx-cosecx+c

**Question 18.**

**Solution:**

**Question 19.**

**Solution:**

put tanx = t

so that sec²x dx = dt

**Question 20.**

**Solution:**

put cosx+sinx=t

⇒(-sinx+cox)dx = dt

**Question 21.**

sin^{-1} (cos x)

**Solution:**

**Question 22.**

**Solution:**

**Question 23.**

(a) tanx+cotx+c

(b) tanx+cosecx+c

(c) -tanx+cotx+c

(d) tanx+secx+c

**Solution:**

(a)

= ∫(sec²x-cosec²x)dx

= tanx+cotx+c

**Question 24.**

(a) -cot(e.x^{x})+c

(b) tan(xe^{x})+c

(c) tan(e^{x})+c

(d) cot e^{x}+c

**Solution:**

(b)

= ∫sec²t dt

= tan t+c = tan(xe^{x})+c

### Chapter 7 Integrals Exercise 7.4

Integrate the functions in exercises 1 to 23

**Question 1.**

**Solution:**

Let x^{3} = t ⇒ 3x²dx = dt

= tan^{-1} (x^{3})+c

**Question 2.**

**Solution:**

**Question 3.**

**Solution:**

put (2-x)=t

so that -dx=dt

⇒ dx=-dt

**Question 4.**

**Solution:**

**Question 5.**

**Solution:**

Put x²=t,so that 2x dx=dt

⇒x dx =

**Question 6.**

**Solution:**

put x^{3} = t,so that 3x²dx = dt

**Question 7.**

**Solution:**

put x²-1 = t,so that 2x dx = dt

**Question 8.**

**Solution:**

put x^{3} = t

so that 3x^{2}dx = dt

**Question 9.**

**Solution:**

let tanx = t

sec x²dx = dt

**Question 10.**

**Solution:**

**Question 11.**

**Solution:**

**Question 12.**

**Solution:**

**Question 13.**

**Solution:**

**Question 14.**

**Solution:**

**Question 15.**

**Solution:**

**Question 16.**

**Solution:**

put 2x²+x-3=t

so that (4x+1)dx=dt

**Question 17.**

**Solution:**

**Question 18.**

**Solution:**

put 5x-2=A(1+2x+3x²)+B

⇒ 6A=5, A=, B=

**Question 19.**

**Solution:**

**Question 20.**

**Solution:**

**Question 21.**

**Solution:**

**Question 22.**

**Solution:**

**Question 23.**

**Solution:**

**Question 24.**

(a) xtan^{-1}(x+1)+c

(b) (x+1)tan^{-1}x+c

(c) tan^{-1}(x+1)+c

(d) tan^{-1}x+c

**Solution:**

(b)

= (x+1)tan^{-1}x+c

**Question 25.**

(a)

(b)

(c)

(d)

**Solution:**

(b)

### Chapter 7 Integrals Exercise 7.5

**Integrate the rational function in exercises 1 to 21**

**Question 1.**

**Solution:**

let ≡

⇒ x ≡ A(x+2)+B(x+1)….(i)

putting x = -1 & x = -2 in (i)

we get A = 1,B = 2

=-log|x+1| + 2log|x+2|+c

**Question 2.**

**Solution:**

let

⇒ x ≡ A(x+3)+B(x-3)…(i)

put x = 3, -3 in (i)

we get &

**Question 3.**

**Solution:**

Let

⇒ 3x-1 = A(x-2)(x-3)+B(x-1)(x-3)+C(x-1)(-2)…..(i)

put x = 1,2,3 in (i)

we get A = 1,B = -5 & C = 4

=log|x-1| – 5log|x-2| + 4log|x+3| + C

**Question 4.**

**Solution:**

let

⇒ x ≡ A(x-2)(x-3)+B(x-1)(x-3)+C(x-1)(x-2)…(i)

put x = 1,2,3 in (i)

**Question 5.**

**Solution:**

let

⇒ 2x = A(x+2)+B(x+1)…(i)

put x = -1, -2 in (i)

we get A = -2, B = 4

=-2log|x+1|+4log|x+2|+c

**Question 6.**

**Solution:**

is an improper fraction therefore we

convert it into a proper fraction. Divide 1 – x² by x – 2x² by long division.

**Question 7.**

**Solution:**

let

⇒ x = A(x²+1)+(Bx+C)(x-1)

Put x = 1,0

⇒

**Question 8.**

**Solution:**

⇒ x ≡ A(x-1)(x+2)+B(x+2)+C(x-1)² …(i)

put x = 1, -2

we get

**Question 9.**

**Solution:**

let

⇒ 3x+5 = A(x-1)(x+1)+B(x+1)+C(x-1)

put x = 1,-1,0

we get

**Question 10.**

**Solution:**

**Question 11.**

**Solution:**

let

**Question 12.**

**Solution:**

**Question 13.**

**Solution:**

⇒ 2 = A(1+x²) + (Bx+C)(1 -x) …(i)

Putting x = 1 in (i), we get; A = 1

Also 0 = A – B and 2 = A + C ⇒B = A = 1 & C = 1

**Question 14.**

**Solution:**

=>3x – 1 = A(x + 2) + B …(i)

Comparing coefficients A = -1 and B = -7

**Question 15.**

**Solution:**

⇒ 1 ≡ A(x-1)(x²+1) + B(x+1)(x²+1) + (Cx+D)(x+1)(x-1) ….(i)

**Question 16.**

[Hint : multiply numerator and denominator by x^{n-1} and put x^{n} = t ]

**Solution:**

**Question 17.**

**Solution:**

put sinx = t

so that cosx dx = dt

**Question 18.**

**Solution:**

put x²=y

**Question 19.**

**Solution:**

put x²=y

so that 2xdx = dy

**Question 20.**

**Solution:**

put x^{4} = t

so that 4x^{3} dx = dt

**Question 21.**

**Solution:**

Let e^{x} = t ⇒ e^{x} dx = dt

⇒

**Question 22.**

choose the correct answer in each of the following :

(a)

(b)

(c)

(d) log|(x-1)(x-2)|+c

**Solution:**

(b)

**Question 23.**

(a)

(b)

(c)

(d)

**Solution:**

(a) let

⇒ 1 = A(x²+1)+(Bx+C)(x)

### Chapter 7 Integrals Exercise 7.6

**Integrate the functions in Exercises 1 to 22.**

**Question 1.**

x sinx

**Solution:**

By part integration

∫x sinx dx = x(-cosx) – ∫1(-cosx)dx

=-x cosx + ∫cosxdx

=-x cosx + sinx + c

**Question 2.**

x sin3x

**Solution:**

∫x sin3x dx =

**Question 3.**

**Solution:**

**Question 4.**

x logx

**Solution:**

**Question 5.**

x log2x

**Solution:**

**Question 6.**

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

**Solution:**

let I =

**Question 10.**

**Solution:**

**Question 11.**

**Solution:**

**Question 12.**

x sec²x

**Solution:**

∫x sec²x dx =x(tanx)-∫1.tanx dx

= x tanx+log cosx+c

**Question 13.**

**Solution:**

**Question 14.**

x(logx)²

**Solution:**

∫x(logx)² dx

**Question 15.**

(x²+1)logx

**Solution:**

∫(x²+1)logx dx

**Question 16.**

**Solution:**

**Question 17.**

**Solution:**

**Question 18.**

**Solution:**

**Question 19.**

**Solution:**

put

**Question 20.**

**Solution:**

**Question 21.**

**Solution:**

let

**Question 22.**

**Solution:**

Put x = tan t

so that dx = sec² t dt

**Choose the correct answer in exercise 23 and 24**

**Question 23.**

(a)

(b)

(c)

(d)

**Solution:**

(a) let x³ = t

⇒3x² dx = dt

**Question 24.**

(a)

(b)

(c)

(d)

**Solution:**

(b)

### Chapter 7 Integrals Exercise 7.7

**Integral the function in exercises 1 to 9**

**Question 1.**

**Solution:**

**Question 2.**

**Solution:**

**Question 3.**

**Solution:**

**Question 4.**

**Solution:**

**Question 5.**

**Solution:**

**Question 6.**

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

**Solution:**

**Choose the correct answer in the Exercises 10 to 11:**

**Question 10.**

(a)

(b)

(c)

(d)

**Solution:**

(a)

**Question 11.**

**Solution:**

(d)

### Chapter 7 Integrals Exercise 7.8

**Evaluate the following definite integral as limit of sums.**

**Question 1.**

**Solution:**

on comparing

we have

**Question 2.**

**Solution:**

on comparing

we have f(x) = x+1, a = 0, b = 5

and nh = b-a = 5-0 = 5

**Question 3.**

**Solution:**

compare

we have

**Question 4.**

**Solution:**

compare

we have f(x) = x²-x and a = 1, b = 4

**Question 5.**

**Solution:**

compare

we have

**Question 6.**

**Solution:**

let f(x) = x + e^{2x},

a = 0, b = 4

and nh = b – a = 4 – 0 = 4

### Chapter 7 Integrals Exercise 7.9

**Evaluate the definite integrals in Exercise 1 to 20.**

**Question 1.**

**Solution:**

**Question 2.**

**Solution:**

**Question 3.**

**Solution:**

**Question 4.**

**Solution:**

**Question 5.**

**Solution:**

**Question 6.**

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

**Solution:**

**Question 10.**

**Solution:**

**Question 11.**

**Solution:**

**Question 12.**

**Solution:**

**Question 13.**

**Solution:**

**Question 14.**

**Solution:**

**Question 15.**

**Solution:**

let x² = t ⇒ 2xdx = dt

when x = 0, t = 0 & when x = 1,t = 1

**Question 16.**

**Solution:**

**Question 17.**

**Solution:**

**Question 18.**

**Solution:**

**Question 19.**

**Solution:**

**Question 20.**

**Solution:**

**Question 21.**

(a)

(b)

(c)

(d)

**Solution:**

(d)

**Question 22.**

(a)

(b)

(c)

(d)

**Solution:**

(c)

### Chapter 7 Integrals Exercise 7.10

**Evaluate the integrals in Exercises 1 to 8 using substitution.**

**Question 1.**

**Solution:**

Let x² + 1 = t

⇒2xdx = dt

when x = 0, t = 1 and when x = 1, t = 2

**Question 2.**

**Solution:**

put sinφ = t,so that cosφdφ = dt

**Question 3.**

**Solution:**

let x = tanθ =>dx = sec²θ dθ

when x = 0 => θ = 0

and when x = 1 =>

**Question 4.**

**Solution:**

let x+2 = t =>dx = dt

when x = 0,t = 2 and when x = 2, t = 4

**Question 5.**

**Solution:**

put cosx = t

so that -sinx dx = dt

when x = 0, t = 1; when , t = 0

**Question 6.**

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

**Solution:**

let 2x = t ⇒ 2dx = dt

when x = 1, t = 2 and when x = 2, t = 4

**Choose the correct answer in Exercises 9 and 10**

**Question 9.**

The value of integral is

(a) 6

(b) 0

(c) 3

(d) 4

**Solution:**

(a) let I =

**Question 10.**

(a) cosx+xsinx

(b) xsinx

(c) xcosx

(d) sinx+xcosx

**Solution:**

(b)

=-x cox+sinx

### Chapter 7 Integrals Exercise 7.11

**By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.**

**Question 1.**

**Solution:**

**Question 2.**

**Solution:**

let I =

**Question 3.**

**Solution:**

let I =

**Question 4.**

**Solution:**

let I =

**Question 5.**

**Solution:**

at x = – 5, x + 2 < 0; at x = – 2, x + 2 = 0; at x = 5, x + 2>0;x + 2<0, x + 2 = 0, x + 2>0

**Question 6.**

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

**Solution:**

let I =

**Question 9.**

**Solution:**

let 2-x = t

⇒ – dx = dt

when x = 0, t = 2 and when x = 2,t = 0

**Question 10.**

**Solution:**

**Question 11.**

**Solution:**

Let f(x) = sin² x

f(-x) = sin² x = f(x)

∴ f(x) is an even function

**Question 12.**

**Solution:**

let I = …(i)

**Question 13.**

**Solution:**

Let f(x) = sin^{7} xdx

⇒ f(-x) = -sin^{7} x = -f(x)

⇒ f(x) is an odd function of x

⇒

**Question 14.**

**Solution:**

let f(x) = cos^{5} x

⇒ f(2π – x) = cos^{5} x

**Question 15.**

**Solution:**

let I = …(i)

**Question 16.**

**Solution:**

let I =

then I =

**Question 17.**

**Solution:**

let I = …(i)

**Question 18.**

**Solution:**

**Question 19.**

show that if f and g are defined as f(x)=f(a-x) and g(x)+g(a-x)=4

**Solution:**

let I =

**Question 20.**

The value of is

(a) 0

(b) 2

(c) π

(d) 1

**Solution:**

(c) let I = is

**Question 21.**

The value of is

(a) 2

(b)

(c) 0

(d) -2

**Solution:**

let I =

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