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NCERT Solutions for 12th Class Maths: Chapter 7-Integrals

Class 12: Maths Chapter 7 solutions. Complete Class 12 Maths Chapter 7 Notes. NCERT Solutions for 12th Class Maths: Chapter 7-Integrals Class 12: Maths Chapter 7 solutions. Complete Class 12 Maths...

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Class 12: Maths Chapter 7 solutions. Complete Class 12 Maths Chapter 7 Notes.

NCERT Solutions for 12th Class Maths: Chapter 7-Integrals

Class 12: Maths Chapter 7 solutions. Complete Class 12 Maths Chapter 7 Notes.

Ex 7.1 Class 12 Maths Question 1.
sin 2x
Solution:

Ex 7.1 Class 12 Maths Question 2.
cos 3x
Solution:

Ex 7.1 Class 12 Maths Question 3.

Solution:

Ex 7.1 Class 12 Maths Question 4.
(ax + c)²
Solution:

Ex 7.1 Class 12 Maths Question 5.

Solution:

Find the following integrals in Exercises 6 to 20 :

Ex 7.1 Class 12 Maths Question 6.

Solution:

Ex 7.1 Class 12 Maths Question 7.

Solution:

Ex 7.1 Class 12 Maths Question 8.

Solution:

Ex 7.1 Class 12 Maths Question 9.

Solution:

Ex 7.1 Class 12 Maths Question 10.

Solution:

Ex 7.1 Class 12 Maths Question 11.

Solution:

Ex 7.1 Class 12 Maths Question 12.

Solution:

Ex 7.1 Class 12 Maths Question 13.

Solution:

Ex 7.1 Class 12 Maths Question 14.

Solution:

Ex 7.1 Class 12 Maths Question 15.

Solution:

Ex 7.1 Class 12 Maths Question 16.

Solution:

Ex 7.1 Class 12 Maths Question 17.

Solution:

Ex 7.1 Class 12 Maths Question 18.

Solution:

= tanx + secx + c

Ex 7.1 Class 12 Maths Question 19.

Solution:

Ex 7.1 Class 12 Maths Question 20.

Solution:

= 2tanx – 3secx + c

Choose the correct answer in Exercises 21 and 22.

Ex 7.1 Class 12 Maths Question 21.
The antiderivative

equals
(a)

(b)

(c)

(d)

Solution:
(c)

Ex 7.1 Class 12 Maths Question 22.
If

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

(b)

(c)

(d)

Solution:
(a)

Ex 7.1 Class 12 Maths Question 1.

Solution:
Let 1+x² = t
⇒ 2xdx = dt

Ex 7.2 Class 12 Maths Question 2.

Solution:
Let logx = t
⇒

Ex 7.2 Class 12 Maths Question 3.

Solution:
Put 1+logx = t
∴

= log|1+logx|+c

Ex 7.2 Class 12 Maths Question 4.
sinx sin(cosx)
Solution:
Put cosx = t, -sinx dx = dt

Ex 7.2 Class 12 Maths Question 5.
sin(ax+b) cos(ax+b)
Solution:
let sin(ax+b) = t
⇒ cos(ax+b)dx = dt

Ex 7.2 Class 12 Maths Question 6.

Solution:

Ex 7.2 Class 12 Maths Question 7.

Solution:
Let x+2 = t²
⇒ dx = 2t dt

Ex 7.2 Class 12 Maths Question 8.

Solution:
let 1+2x² = t²
⇒ 4x dx = 2t dt

Ex 7.2 Class 12 Maths Question 9.

Solution:
let x²+x+1 = t
⇒(2x+1)dx = dt

Ex 7.2 Class 12 Maths Question 10.

Solution:

Let √x-1 = t

= 2logt + c
= 2log(√x-1)+c

Ex 7.2 Class 12 Maths Question 11.

Solution:
let x+4 = t
⇒ dx = dt, x = t-4

Ex 7.2 Class 12 Maths Question 12.

Solution:

Ex 7.2 Class 12 Maths Question 13.

Solution:
Let 2+3x³ = t
⇒ 9x² dx = dt

Ex 7.2 Class 12 Maths Question 14.

Solution:
Put log x = t, so that

Ex 7.2 Class 12 Maths Question 15.

Solution:
put 9-4x² = t, so that -8x dx = dt

Ex 7.2 Class 12 Maths Question 16.

Solution:
put 2x+3 = t
so that 2dx = dt

Ex 7.2 Class 12 Maths Question 17.

Solution:
Let x² = t
⇒ 2xdx = dt ⇒

Ex 7.2 Class 12 Maths Question 18.

Solution:

Ex 7.2 Class 12 Maths Question 19.

Solution:

put e^x+e^-x = t
so that (e^x-e^-x)dx = dt

Ex 7.2 Class 12 Maths Question 20.

Solution:
put e^2x-e^-2x = t
so that (2e^2x-2e^-2x)dx = dt

Ex 7.2 Class 12 Maths 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
=

Ex 7.2 Class 12 Maths Question 22.
sec²(7-4x)
Solution:
∫sec²(7-4x)dx
=

Ex 7.2 Class 12 Maths Question 23.

Solution:

Ex 7.2 Class 12 Maths Question 24.

Solution:
put 2sinx+4cosx = t
⇒ (2cosx-3sinx)dx = dt

Ex 7.2 Class 12 Maths Question 25.

Solution:
put 1-tanx = t
so that -sec²x dx = dt

Ex 7.2 Class 12 Maths Question 26.

Solution:

= 2sin√x+c

Ex 7.2 Class 12 Maths Question 27.

Solution:
put sin2x = t²
⇒ cos2x dx = t dt

Ex 7.2 Class 12 Maths Question 28.

Solution:
put 1+sinx = t²
⇒cosx dx = 2t dt

Ex 7.2 Class 12 Maths Question 29.
cotx log sinx
Solution:
put log sinx = t,
⇒ cot x dx = dt

Ex 7.2 Class 12 Maths Question 30.

Solution:
put 1+cosx = t
⇒ -sinx dx = dt

=-log(1+cosx)+c

Ex 7.2 Class 12 Maths Question 31.

Solution:
put 1+cosx = t
so that -sinx dx = dt

Ex 7.2 Class 12 Maths Question 32.

Solution:

Ex 7.2 Class 12 Maths Question 33.

Solution:

Ex 7.2 Class 12 Maths Question 34.

Solution:

Ex 7.2 Class 12 Maths Question 35.

Solution:
let 1+logx = t
⇒

Ex 7.2 Class 12 Maths Question 36.

Solution:
put x+logx = t

Ex 7.2 Class 12 Maths Question 37.

Solution:

Choose the correct answer in exercises 38 and 39

Ex 7.2 Class 12 Maths Question 38.

(a) 10^x – x¹⁰ + C
(b) 10^x + x¹⁰ + C
(c) (10^x – x¹⁰) + C
(d) log (10^x + x¹⁰) + C
Solution:
(d)

= log (10^x + x¹⁰) + C

Ex 7.2 Class 12 Maths Question 39.

(a) tanx + cotx + c
(b) tanx – cotx + c
(c) tanx cotx + c
(d) tanx – cot2x + c
Solution:
(c)

= tanx – cotx + c

Ex 7.3 Class 12 Maths Question 1.
sin²(2x+5)
Solution:
∫sin²(2x+5)dx
= $\frac { 1 }{ 2 }$ ∫[1-cos2(2x+5)]dx
= $\frac { 1 }{ 2 }$ ∫[1-cos(4x+10)]dx
=

Ex 7.3 Class 12 Maths Question 2.
sin3x cos4x
Solution:
∫sin3x cos4x
= $\frac { 1 }{ 2 }$ ∫[sin(3x+4x)+cos(3x-4x)]dx
= $\frac { 1 }{ 2 }$ ∫[sin7x+sin(-x)]dx
=

Ex 7.3 Class 12 Maths Question 3.
∫cos2x cos4x cos6x dx
Solution:
$\frac { 1 }{ 2 }$ ∫cos2x cos4x cos6x dx
= $\frac { 1 }{ 2 }$ ∫(cos6x+cos2x) cos6x dx

Ex 7.3 Class 12 Maths Question 4.
∫sin³(2x+1)dx
Solution:
= $\frac { 1 }{ 4 }$ ∫[3sin(2x+1)-sin3(2x+1)]dx
=

Ex 7.3 Class 12 Maths Question 5.
sin³x cos³x
Solution:
put sin x = t
⇒ cos x dx = dt

Ex 7.3 Class 12 Maths Question 6.
sinx sin2x sin3x
Solution:
∫sinx sin2x sin3x dx
= $\frac { 1 }{ 2 }$ ∫ 2sin x sin 2x sin 3x dx
= $\frac { 1 }{ 2 }$ ∫ (cosx – cos3x)sin 3x dx
= $\frac { 1 }{ 2 }$ ∫ (sin 4x + sin 2x – sin 6x)dx
=

Ex 7.3 Class 12 Maths Question 7.
sin 4x sin 8x
Solution:

∫sin 4x sin 8xdx
=

∫(cos 4x – cos 12x)dx
=

Ex 7.3 Class 12 Maths Question 8.

Solution:

Ex 7.3 Class 12 Maths Question 9.

Solution:

Ex 7.3 Class 12 Maths Question 10.
∫sinx⁴ dx
Solution:

Ex 7.3 Class 12 Maths Question 11.
cos⁴ 2x
Solution:
∫ cos⁴ 2x dx

Ex 7.3 Class 12 Maths Question 12.

Solution:

Ex 7.3 Class 12 Maths Question 13.

Solution:
let I =

= 2∫cos x dx + 2cos α∫dx
= 2(sinx+xcosα)+c

Ex 7.3 Class 12 Maths Question 14.

Solution:
let I =

put cosx+sinx = t
⇒ (-sinx+cosx)dx = dt

Ex 7.3 Class 12 Maths Question 15.

Solution:
I = ∫(sec²2x-1)sec2x tan 2xdx
put sec2x=t,2 sec2x tan2x dx=dt

Ex 7.3 Class 12 Maths Question 16.
tan⁴x
Solution:
let I = ∫tan⁴ dx
= ∫(sec²x-1)²dx

Ex 7.3 Class 12 Maths Question 17.

Solution:

= secx-cosecx+c

Ex 7.3 Class 12 Maths Question 18.

Solution:

Ex 7.3 Class 12 Maths Question 19.

Solution:

put tanx = t
so that sec²x dx = dt

Ex 7.3 Class 12 Maths Question 20.

Solution:

put cosx+sinx=t
⇒(-sinx+cox)dx = dt

Ex 7.3 Class 12 Maths Question 21.
sin^-1 (cos x)
Solution:

Ex 7.3 Class 12 Maths Question 22.

Solution:

Ex 7.3 Class 12 Maths 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

Ex 7.3 Class 12 Maths 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

Ex 7.4 Class 12 Maths Question 1.

Solution:
Let x³ = t ⇒ 3x²dx = dt

= tan^-1 (x³)+c

Ex 7.4 Class 12 Maths Question 2.

Solution:

Ex 7.4 Class 12 Maths Question 3.

Solution:
put (2-x)=t
so that -dx=dt
⇒ dx=-dt

Ex 7.4 Class 12 Maths Question 4.

Solution:

Ex 7.4 Class 12 Maths Question 5.

Solution:
Put x²=t,so that 2x dx=dt
⇒x dx =

Ex 7.4 Class 12 Maths Question 6.

Solution:
put x³ = t,so that 3x²dx = dt

Ex 7.4 Class 12 Maths Question 7.

Solution:

put x²-1 = t,so that 2x dx = dt

Ex 7.4 Class 12 Maths Question 8.

Solution:
put x³ = t
so that 3x²dx = dt

Ex 7.4 Class 12 Maths Question 9.

Solution:
let tanx = t
sec x²dx = dt

Ex 7.4 Class 12 Maths Question 10.

Solution:

Ex 7.4 Class 12 Maths Question 11.

Solution:

Ex 7.4 Class 12 Maths Question 12.

Solution:

Ex 7.4 Class 12 Maths Question 13.

Solution:

Ex 7.4 Class 12 Maths Question 14.

Solution:

Ex 7.4 Class 12 Maths Question 15.

Solution:

Ex 7.4 Class 12 Maths Question 16.

Solution:

put 2x²+x-3=t
so that (4x+1)dx=dt

Ex 7.4 Class 12 Maths Question 17.

Solution:

Ex 7.4 Class 12 Maths Question 18.

Solution:
put 5x-2=A

(1+2x+3x²)+B
⇒ 6A=5, A=

, B=

Ex 7.4 Class 12 Maths Question 19.

Solution:

Ex 7.4 Class 12 Maths Question 20.

Solution:

Ex 7.4 Class 12 Maths Question 21.

Solution:

Ex 7.4 Class 12 Maths Question 22.

Solution:

Ex 7.4 Class 12 Maths Question 23.

Solution:

Ex 7.4 Class 12 Maths Question 24.

(a) xtan^-1(x+1)+c
(b) (x+1)tan^-1x+c
(c) tan^-1(x+1)+c
(d) tan^-1x+c
Solution:
(b)

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

Ex 7.4 Class 12 Maths Question 25.

(a)

(b)

(c)

(d)

Solution:
(b)

Ex 7.5 Class 12 Maths 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

Ex 7.5 Class 12 Maths Question 2.

Solution:
let

⇒ x ≡ A(x+3)+B(x-3)…(i)
put x = 3, -3 in (i)
we get

Ex 7.5 Class 12 Maths 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

Ex 7.5 Class 12 Maths 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)

Ex 7.5 Class 12 Maths 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

Ex 7.5 Class 12 Maths Question 6.

Solution:

is an improper fraction therefore we
convert it into a proper fraction. Divide 1 – x² by x – 2x² by long division.

Ex 7.5 Class 12 Maths Question 7.

Solution:
let

⇒ x = A(x²+1)+(Bx+C)(x-1)
Put x = 1,0
⇒

Ex 7.5 Class 12 Maths Question 8.

Solution:

⇒ x ≡ A(x-1)(x+2)+B(x+2)+C(x-1)² …(i)
put x = 1, -2
we get

Ex 7.5 Class 12 Maths Question 9.

Solution:
let

⇒ 3x+5 = A(x-1)(x+1)+B(x+1)+C(x-1)
put x = 1,-1,0
we get

Ex 7.5 Class 12 Maths Question 10.

Solution:

Ex 7.5 Class 12 Maths Question 11.

Solution:
let

Ex 7.5 Class 12 Maths Question 12.

Solution:

Ex 7.5 Class 12 Maths 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

Ex 7.5 Class 12 Maths Question 14.

Solution:

=>3x – 1 = A(x + 2) + B …(i)
Comparing coefficients A = -1 and B = -7

$=3log|x+2|+\frac { 7 }{ x+2 } +c$

Ex 7.5 Class 12 Maths Question 15.

Solution:

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

Ex 7.5 Class 12 Maths Question 16.

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

Ex 7.5 Class 12 Maths Question 17.

Solution:
put sinx = t
so that cosx dx = dt

Ex 7.5 Class 12 Maths Question 18.

Solution:
put x²=y

Ex 7.5 Class 12 Maths Question 19.

Solution:
put x²=y
so that 2xdx = dy

Ex 7.5 Class 12 Maths Question 20.

Solution:
put x⁴ = t
so that 4x³ dx = dt

Ex 7.5 Class 12 Maths Question 21.

Solution:
Let e^x = t ⇒ e^x dx = dt
⇒

Ex 7.5 Class 12 Maths Question 22.
choose the correct answer in each of the following :

(a)

(b)

(c)

(d) log|(x-1)(x-2)|+c
Solution:
(b)

Ex 7.5 Class 12 Maths Question 23.

(a)

(b)

(c)

(d) $\frac { 1 }{ 2 } log|x|+log({ x }^{ 2 }+1)+c$
Solution:
(a) let

⇒ 1 = A(x²+1)+(Bx+C)(x)

Ex 7.6 Class 12 Maths Question 1.
x sinx
Solution:
By part integration
∫x sinx dx = x(-cosx) – ∫1(-cosx)dx
=-x cosx + ∫cosxdx
=-x cosx + sinx + c

Ex 7.6 Class 12 Maths Question 2.
x sin3x
Solution:
∫x sin3x dx =

Ex 7.6 Class 12 Maths Question 3.

Solution:

Ex 7.6 Class 12 Maths Question 4.
x logx
Solution:

Ex 7.6 Class 12 Maths Question 5.
x log2x
Solution:

Ex 7.6 Class 12 Maths Question 6.
${ x }^{ 2 }logx$
Solution:

Ex 7.6 Class 12 Maths Question 7.

Solution:

Ex 7.6 Class 12 Maths Question 8.

Solution:

Ex 7.6 Class 12 Maths Question 9.

Solution:
let I =

Ex 7.6 Class 12 Maths Question 10.

Solution:

Ex 7.6 Class 12 Maths Question 11.

Solution:

Ex 7.6 Class 12 Maths Question 12.
x sec²x
Solution:
∫x sec²x dx =x(tanx)-∫1.tanx dx
= x tanx+log cosx+c

Ex 7.6 Class 12 Maths Question 13.

Solution:

Ex 7.6 Class 12 Maths Question 14.
x(logx)²
Solution:
∫x(logx)² dx

Ex 7.6 Class 12 Maths Question 15.
(x²+1)logx
Solution:
∫(x²+1)logx dx

Ex 7.6 Class 12 Maths Question 16.

Solution:

Ex 7.6 Class 12 Maths Question 17.

Solution:

Ex 7.6 Class 12 Maths Question 18.

Solution:

Ex 7.6 Class 12 Maths Question 19.

Solution:
put

Ex 7.6 Class 12 Maths Question 20.

Solution:

Ex 7.6 Class 12 Maths Question 21.

Solution:
let

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 Q21.1

Ex 7.6 Class 12 Maths Question 22.
${ sin }^{ -1 }\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right)$
Solution:
Put x = tan t
so that dx = sec² t dt

Choose the correct answer in exercise 23 and 24

Ex 7.6 Class 12 Maths Question 23.

(a)

(b)

(c)

(d)

Solution:
(a) let x³ = t
⇒3x² dx = dt

Ex 7.6 Class 12 Maths Question 24.

(a)

(b)

(c)

(d)

Solution:
(b)

Ex 7.7 Class 12 Maths Question 1.

Solution:

Ex 7.7 Class 12 Maths Question 2.

Solution:

Ex 7.7 Class 12 Maths Question 3.

Solution:

Ex 7.7 Class 12 Maths Question 4.

Solution:

Ex 7.7 Class 12 Maths Question 5.

Solution:

Ex 7.7 Class 12 Maths Question 6.

Solution:

Ex 7.7 Class 12 Maths Question 7.

Solution:

Ex 7.7 Class 12 Maths Question 8.

Solution:

Ex 7.7 Class 12 Maths Question 9.

Solution:

Choose the correct answer in the Exercises 10 to 11:

Ex 7.7 Class 12 Maths Question 10.

(a)

(b)

(c)

(d)

Solution:
(a)

Ex 7.7 Class 12 Maths Question 11.

Solution:
(d)

Ex 7.8 Class 12 Maths Question 1.

Solution:
on comparing

we have

Ex 7.8 Class 12 Maths Question 2.

Solution:
on comparing

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

Ex 7.8 Class 12 Maths Question 3.

Solution:
compare

we have

Ex 7.8 Class 12 Maths Question 4.

Solution:
compare

we have f(x) = x²-x and a = 1, b = 4

Ex 7.8 Class 12 Maths Question 5.

Solution:
compare

we have

Ex 7.8 Class 12 Maths Question 6.

Solution:
let f(x) = x + e^2x,
a = 0, b = 4
and nh = b – a = 4 – 0 = 4

Ex 7.9 Class 12 Maths Question 1.

Solution:

Ex 7.9 Class 12 Maths Question 2.

Solution:

Ex 7.9 Class 12 Maths Question 3.

Solution:

Ex 7.9 Class 12 Maths Question 4.

Solution:

Ex 7.9 Class 12 Maths Question 5.

Solution:

Ex 7.9 Class 12 Maths Question 6.

Solution:

Ex 7.9 Class 12 Maths Question 7.

Solution:

Ex 7.9 Class 12 Maths Question 8.

Solution:

Ex 7.9 Class 12 Maths Question 9.

Solution:

Ex 7.9 Class 12 Maths Question 10.

Solution:

Ex 7.9 Class 12 Maths Question 11.

Solution:

Ex 7.9 Class 12 Maths Question 12.

Solution:

Ex 7.9 Class 12 Maths Question 13.

Solution:

Ex 7.9 Class 12 Maths Question 14.

Solution:

Ex 7.9 Class 12 Maths Question 15.

Solution:
let x² = t ⇒ 2xdx = dt
when x = 0, t = 0 & when x = 1,t = 1

Ex 7.9 Class 12 Maths Question 16.

Solution:

Ex 7.9 Class 12 Maths Question 17.

Solution:

Ex 7.9 Class 12 Maths Question 18.

Solution:

Ex 7.9 Class 12 Maths Question 19.

Solution:

Ex 7.9 Class 12 Maths Question 20.

Solution:

Ex 7.9 Class 12 Maths Question 21.

(a)

(b)

(c)

(d)

Solution:
(d)

Ex 7.9 Class 12 Maths Question 22.

(a)

(b)

(c)

(d)

Solution:
(c)

Ex 7.10 Class 12 Maths Question 1.

Solution:
Let x² + 1 = t
⇒2xdx = dt
when x = 0, t = 1 and when x = 1, t = 2

Ex 7.10 Class 12 Maths Question 2.

Solution:

put sinφ = t,so that cosφdφ = dt

Ex 7.10 Class 12 Maths Question 3.

Solution:
let x = tanθ =>dx = sec²θ dθ
when x = 0 => θ = 0
and when x = 1 => $\theta \frac { \pi }{ 4 }$
$\frac { 1 }{ 2 }$

Ex 7.10 Class 12 Maths Question 4.

Solution:
let x+2 = t =>dx = dt
when x = 0,t = 2 and when x = 2, t = 4

Ex 7.10 Class 12 Maths Question 5.

Solution:
put cosx = t
so that -sinx dx = dt
when x = 0, t = 1; when

, t = 0

Ex 7.10 Class 12 Maths Question 6.

Solution:

Ex 7.10 Class 12 Maths Question 7.

Solution:

Ex 7.10 Class 12 Maths 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

Ex 7.10 Class 12 Maths Question 9.
The value of integral

is
(a) 6
(b) 0
(c) 3
(d) 4
Solution:
(a) let I =

Ex 7.10 Class 12 Maths Question 10.

(a) cosx+xsinx
(b) xsinx
(c) xcosx
(d) sinx+xcosx
Solution:
(b)

=-x cox+sinx

Ex 7.11 Class 12 Maths Question 1.

Solution:

Ex 7.11 Class 12 Maths Question 2.

Solution:
let I =

Ex 7.11 Class 12 Maths Question 3.

Solution:
let I =

Ex 7.11 Class 12 Maths Question 4.

Solution:
let I =

Ex 7.11 Class 12 Maths 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

Ex 7.11 Class 12 Maths Question 6.

Solution:

Ex 7.11 Class 12 Maths Question 7.

Solution:

Ex 7.11 Class 12 Maths Question 8.

Solution:
let I =

Ex 7.11 Class 12 Maths Question 9.

Solution:
let 2-x = t
⇒ – dx = dt
when x = 0, t = 2 and when x = 2,t = 0

Ex 7.11 Class 12 Maths Question 10.

Solution:

Ex 7.11 Class 12 Maths Question 11.

Solution:
Let f(x) = sin² x
f(-x) = sin² x = f(x)
∴ f(x) is an even function

Ex 7.11 Class 12 Maths Question 12.

Solution:
let I =

…(i)

Ex 7.11 Class 12 Maths Question 13.

Solution:
Let f(x) = sin⁷ xdx
⇒ f(-x) = -sin⁷ x = -f(x)
⇒ f(x) is an odd function of x
⇒

Ex 7.11 Class 12 Maths Question 14.

Solution:
let f(x) = cos⁵ x
⇒ f(2π – x) = cos⁵ x

Ex 7.11 Class 12 Maths Question 15.

Solution:
let I =

…(i)

Ex 7.11 Class 12 Maths Question 16.

Solution:
let I =

then I =

Ex 7.11 Class 12 Maths Question 17.

Solution:
let I =

…(i)

Ex 7.11 Class 12 Maths Question 18.

Solution:

Ex 7.11 Class 12 Maths 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 =

Ex 7.11 Class 12 Maths Question 20.
The value of

is
(a) 0
(b) 2
(c) π
(d) 1
Solution:
(c) let I =

Ex 7.11 Class 12 Maths Question 21.
The value of

is
(a) 2
(b)

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NCERT Solutions for 12th Class Maths: Chapter 7-Integrals

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NCERT Solutions for 12th Class Maths: Chapter 7-Integrals

NCERT Solutions for 12th Class Maths: Chapter 7: Download PDF

Chapterwise NCERT Solutions for Class 12 Maths :

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