Class 12: Maths Chapter 7 solutions. Complete Class 12Maths Chapter 7 Notes.
Contents
NCERT Solutions for 12th Class Maths: Chapter 7-Integrals
Class 12:Maths Chapter 7 solutions. Complete Class 12Maths 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 ex+e-x = t
so that (ex-e-x)dx = dt
Ex 7.2 Class 12 Maths Question 20.
Solution:
put e2x-e-2x = t
so that (2e2x-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) 10x – x10 + C
(b) 10x + x10 + C
(c) (10x – x10) + C
(d) log (10x + x10) + C
Solution:
(d)
= log (10x + 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
= ∫[1-cos2(2x+5)]dx
=
=
Ex 7.3 Class 12 Maths Question 2.
sin3x cos4x
Solution:
∫sin3x cos4x
=
=
=
Ex 7.3 Class 12 Maths Question 3.
∫cos2x cos4x cos6x dx
Solution:
=
Ex 7.3 Class 12 Maths Question 4.
∫sin3(2x+1)dx
Solution:
=
=
=
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
=
=
=
=
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.xx)+c
(b) tan(xex)+c
(c) tan(ex)+c
(d) cot ex+c
Solution:
(b)
= ∫sec²t dt
= tan t+c = tan(xex)+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
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 xn-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 ex = t ⇒ ex 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)
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.
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
Ex 7.6 Class 12 Maths Question 22.
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 + e2x,
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 =>
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 =
is
Ex 7.11 Class 12 Maths Question 21.
The value of
is
(a) 2
(b)
(c) 0
(d) -2
Solution:
let I =
NCERT Solutions for 12th Class Maths: Chapter 7: Download PDF
NCERT Solutions for 12th Class Maths: Chapter 7-Integrals
Download PDF: NCERT Solutions for 12th Class Maths: Chapter 7-Integrals PDF
Chapterwise NCERT Solutions for Class 12 Maths :
- Chapter 1 – Relations and Functions
- Chapter 2 – Inverse Trigonometric Functions.
- Chapter 3 – Matrices
- Chapter 4 – Determinants.
- Chapter 5 – Continuity and Differentiability.0.0
- Chapter 6 – Application of Derivatives.
- Chapter 7 – Integrals.
- Chapter 8 – Application of Integrals.
- Chapter 9: Differential Equations
- Chapter 10: Vector Algebra
- Chapter 11: Three Dimensional Geometry
- Chapter 12: Linear Programming
- Chapter 13: Probability
About NCERT
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