GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6

   

Gujarat Board GSEB Textbook Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 11 Maths Chapter 7 Integrals Ex 7.6

Integrate the following functions:
Question 1.
x sin x
Solution:
Let I = ∫x sin x dx
We know that ∫uv dx = u∫v dx – ∫(u’ ∫v dx)dx
Put u = x and v = sin x.
∴ I = x ∫sin x dx = ∫[(\(\frac{d}{dx}\) x) ∫sin x dx]dx
= – x cos x – ∫1. (- cos x)dx
= – x cos x + ∫cos x dx = – x cos x + sin x + C.

GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 2.
x sin 3x
Solution:
Let I = ∫xsin 3x dx. Taking u = x and v = sin 3x.
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 1

Question 3.
x2ex
Solution:
Let I = ∫x2ex dx, Put u = x2 and v = exdx.
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 2
Taking x as I function, integrating again, we get
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 3

GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 4.
x log x
Solution:
Let I = ∫xlog x dx, Put u = log x and v = x.
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 4

Question 5.
x log 2x
Solution:
Let I = ∫xlog 2x dx = ∫(log 2x).x dx
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 5

Question 6.
x2log x
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 6

GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 7.
x sin-1x
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 7
Put x = sinθ so that dx = cosθdθ.
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 8

Question 8.
x tan-1x
Solution:
Let I = ∫x tan-1 x dx. Put x = tan-1x and v = x.
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 9

Question 9.
x cos-1x
Solution:
Let I = ∫xcos-1 x dx = ∫cos-1 x.x dx
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 10
Putting x = cosθ so that dx = – sinθ dθ.
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 11

GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 10.
(sin-1x)2
Solution:
Let I = ∫(sin-1 x2)dx.
Put sin-1x = θ ⇒ x = sin θ ⇒ dx = cosθ dθ.
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 12

Question 11.
\(\frac{x \cos ^{-1} x}{\sqrt{1-x^{2}}}\)
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 13

Question 12.
x sec2x
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 14

Question 13.
tan-1x
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 15

GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 14.
x(log x)2
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 16

Question 15.
(x2 + 1) log x
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 17

Question 16.
ex(sin x + cos x)
Solution:
Let I = ∫ex(sin x + cos x) dx.
Put exsin x = t so that (ex.cos x + sin x.ex)dx = dt.
⇒ ex(sin x + cos x) dx = dt
∴ I = ∫dt = t + C = exsinx + C.

GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 17.
\(\frac{x e^{x}}{(1+x)^{2}}\)
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 18

Question 18.
ex(\(\frac{1+sinx}{1+cosx}\))
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 19
Taking tan \(\frac{x}{2}\) as I function and ex as II function,
integrating ∫ex tan \(\frac{x}{2}\) dx by parts, we get
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 20

Question 19.
ex(\(\frac{1}{x}\) – \(\frac{1}{x^{2}}\))
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 21

GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 20.
\(\frac{(x-3) e^{x}}{(x-1)^{3}}\)
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 22

Question 21.
e2xsinx
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 23

Question 22.
sin-1(\(\frac{2 x}{1+x^{2}}\))
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 24

GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6

Choose the correct answers in the following questions from 23 and 24:
Question 23.
∫ x2 ex3 dx equals
(A) \(\frac{1}{3}\) ex3 + C
(B) \(\frac{1}{3}\) ex2 + C
(C) \(\frac{1}{2}\) ex3 + C
(D) \(\frac{1}{2}\) ex2 + C
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 25
∴ Part (A) is the correct answer.

GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 24.
∫ ex secx(1 + tan x) dx equals
(A) ex cosx + C
(B) ex secx + C
(C) ex sinx + C
(D) ex tanx + C
Solution:
GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.6 img 26
Taking sec x as I function and ex as II function, integrating by parts
I1 = (sec x) ∫ex dx – ∫(sec x tan x ∫ex dx)dx
= (sec x)ex – ∫ex sec x tan x dx
Putting this value in (1), we get
I = I1 + ∫ex sec x tan x dx
= (sec x)ex – ∫ex sec x tan x dx
Putting this value in (1), we get
I = I1 + ∫ex sec x tan x dx
= (sec x)ex – ∫ex sec x tan x dx + ∫ex sec x tan x dx + C
= ex sec x + C.
∴ Part (B) is the correct answer.

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