2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Students can Download Basic Maths Question Bank Chapter 20 Indefinite Integrals Questions and Answers, Notes Pdf, 2nd PUC Basic Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and to clear all their doubts, score well in final exams.

Karnataka 2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

2nd PUC Basic Maths Indefinite Integrals One Mark Questions and Answers

Question 1.
∫(7x – 3)4 dx
Answer:
\(=\frac{(7 x-3)^{5}}{35}+\mathrm{C}\)

Question 2.
∫(2x + 5)3/5 dx
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 1

Question 3.
\(\int \frac{1}{10 x+3} d x\)
Answer:
\(=\frac{\log (10 x+3)}{10}+C\)

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 4.
∫e3 – 4x dx
Answer:
\(\frac{e^{3-4 x}}{-4}+C\)

Question 5.
∫35x – 3
Answer:
\(\frac{3^{5 x-3}}{5 . \log 3}+\mathrm{C}\)

Question 6.
∫sec2(x – 5) dx
Answer:
= tan(x – 5) + C

Question 7.
∫cosec(3 – 5x).cot(3 – 5x) dx
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 2

Question 8.
\(\int \frac{5}{\sec x} d x\)
Answer:
sec x
= 5∫cos x = 5. sin x + C

Question 9.
\(\int \frac{8}{\sin ^{2} x} d x\)
Answer:
= ∫8.cosec2 x dx ,
= -8 cot x + C

Question 10.
∫tan2 x dx
Answer:
∫(sec2 x – 1)dx = tan x – x + C

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 11.
Integrate \(\frac{x^{2}-1}{x}\) w.r.t x
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 3

Question 12.
Integrate \(\frac{1}{3-4 x}\) w.r.t x
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 4

Question 13.
Integrate \(\frac{x+5}{x+2}\) w.r.t x
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 5
= x + 3 log (x + 2) + C

Question 14.
∫x. √xdx = ∫x3/2 dx
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 6

Question 15.
\(\int \frac{x+1}{x} d x\)
Answer:
= ∫1 + \(\frac{1}{x}\) dx = x + log x + C

Question 16.
\(\int \sqrt{a x+b} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 7

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 17.
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 8
Answer:
= log (ex + 1) + C

Question 18.
\(\int \frac{1}{2 \sqrt{x}} d x\)
Answer:
= √x + C

Question 19.
Evaluate \(\int \frac{x}{1+x^{2}} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 9

Question 20.
Evaluate \(\int \frac{1}{x}-\frac{1}{x^{2}} d x\)
Answer:
= log x + \(\frac{1}{x}\) + C

Question 21.
Evaluate \(\int \frac{x}{x+4} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 10
= x – 4 log(x + C) + C

Question 22.
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 11
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 12

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 23.
Evaluate ∫(Xe + ex + ee) dx
Answer:
= \(\frac{x^{e+1}}{e+1}\) + ex + eex + C

Question 24.
Evaluate \(\int\left(x+\frac{1}{x}\right) d x\)
Answer:
= \(\frac{x^{2}}{2}\) + log x + C

Question 25.
Evaluate \(\int \frac{1}{e^{x}} d x\)
Answer:
= ∫e-x dx = -e-x + C

Question 26.
Evaluate \(\int(\sqrt{x}+x+1) d x\)
Answer:
\(=\frac{2}{3} x^{\frac{3}{2}}+\frac{x^{2}}{2}+x+C\)

Question 27.
Evaluate \(\int \sqrt{3 x-5} d x\)
Answer:
\(=\frac{2(3 x-5)^{\frac{3}{2}}}{9}+C\)

Question 28.
Evaluate ∫e-5x dx
Answer:
\(=\frac{e^{-5 x}}{-5}+C\)

Question 29.
\(\int \frac{1}{x \log x} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 13

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 30.
Evaluate \(\int \frac{1}{(4 x+1)^{3}} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 14

Question 31.
Evaluate \(\int \frac{\sin x-\cos x}{\sin x+\cos x} d x\)
Answer:
log (sin x + cos x) + C

Question 32.
Evaluate ∫(x2 + 2x – 5)4 (x + 1)dx
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 15

2nd PUC Basic Maths Indefinite Integrals Two or Three Marks Questions and Answers

Question 1.
∫sin 3x . cos x dx
Answer:
By transformation formula
= ∫\(\frac { 1 }{ 2 }\)(sin 4x + sin 2x)dx
= \(\frac{1}{2}\left[\frac{-\cos 4 x}{4}-\frac{\cos 2 x}{2}\right]+C\)

Question 2.
∫sin 4x . sin 2x dx
Answer:
= ∫\(\frac { 1 }{ 2 }\)[cos 2x – cos 6x] dx
\(\frac{1}{2}\left[\frac{\sin 2 x}{2}-\frac{\sin 6 x}{6}\right]+C\)

Question 3.
∫cos 3x . cos 2x dx
Answer:
= ∫\(\frac { 1 }{ 2 }\)[cos(5x) + cos x] dx
= \(\frac{1}{2}\left[\frac{\sin 5 x}{5}+\sin x\right]+C\)

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 4.
∫cos 5x . sin x dx
Answer:
∫\(\frac { 1 }{ 2 }\) [sin 6x – sin 4x]dx
\(\frac{1}{2}\left[-\frac{\cos 6 x}{6}+\frac{\cos 4 x}{4}\right]+C\)

Question 5.
∫cos2 3x dx
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 16

Question 6.
∫sin2 xdx
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 17

Question 7.
∫sin x . cos x dx
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 18

Question 8.
∫sin2 2x . cos2 2x dx
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 19

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 9.
∫sin3 x dx
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 20

Question 10.
∫ cos2 x dx
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 21

Question 11.
\(\int \frac{\sin ^{2} x}{1+\cos x}\)
Answer:
= \(\int \frac{1-\cos ^{2} x}{1+\cos ^{2} x} d x\)
= ∫1 + cos x dx = x + sin x + C

Question 12.
\(\int \frac{1}{1+\cos x} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 22
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 23

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 13.
\(\int \frac{1}{1-\cos 2 x} d x\)
Answer:
\(=\int \frac{1}{2 \sin ^{2} x} d x\)
= \(\frac { 1 }{ 2 }\) ∫cosec2 x dx
= \(\frac { 1 }{ 2 }\) (-cot x) + C

Question 14.
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 24
Answer:
= ∫cosec x\(\sqrt{\cot ^{2} x} d x\)
= ∫cosec x cot x dx = -cosecx+ c

Question 15.
\(\int \sqrt{1+\sin 2 x} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 25
= -cos x + sin x + c.

Question 16.
\(\int \frac{1-\cos 2 x}{1+\cos 2 x} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 26
= ∫(sec2x – 1)dx
= tanx – x + c.

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 17.
\(\int \frac{1}{1+\sin x} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 27
= ∫sec2x – ∫tan x sec x
= tan x – sec x + c

Question 18.
\(\int \frac{1+\sin x}{\cos ^{2} x} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 28
= ∫sec2 x + sec x . tan x dx
= tan x – sec x + cos

Question 19.
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 29
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 30

Question 20.
\(\int \frac{4}{\sqrt{x+1}+\sqrt{x+2}} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 31
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 32

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 21.
\(\int \frac{7 x^{6}+7^{x} \log 7}{x^{7}+7^{x}} d x\)
Answer:
= log (x7 + 7x) + C

Question 22.
\(\int \frac{3^{x} \log 3}{3^{x}+1} d x\)
Answer:
= log (3x + 1)

Question 23.
\(\int \sqrt{x}\left(1-\frac{1}{x}\right) d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 33

Question 24.
\(\int \frac{d x}{1+e^{-x}}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 34
= log (ex + 1) + C

Question 25.
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 35
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 36

Question 26.
\(\int \frac{d x}{1+e^{x}} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 37

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 27.
\(\int \frac{1}{1+2 e^{x}} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 38
= – log(e-x + 2) + C

Question 28.
∫xe5x dx
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 39

Question 29.
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 40
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 41

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 30.
\(\int \frac{\sqrt{4+\log x}}{x} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 42

Question 31.
\(\int \frac{2 x^{2}+5 x+1}{x} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 43

Question 32.
∫ex3 . 3x2 dx
Answer:
∫et . dt
et + c
= ex3 + c
put x3 = t
3x2 dx = dt

Question 33.
∫x log x dx
Answer:
∫log x . dx
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 44

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

2nd PUC Basic Maths Indefinite Integrals Four or Five Marks Questions and Answers

Question 1.
\(\int \frac{1}{\sqrt{x+1}-\sqrt{x-2}} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 45
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 46

Question 2.
\(\int \frac{3 x-1}{(x-1)(x-2)(x-3)} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 47
3x – 1 = A(x – 2)(x – 3) + B(x – 1)(x – 3) + C(x – 1)(x – 2)
put x = 1
3 – 1 = A(1 – 2)(1 – 3) + B(0) + C(0)
2 = A(-1) (-2) ⇒ A = \(\frac{2}{2}\) = 1
put x = 2
6 – 1 = A(0) + B(2 – 1)(2 – 3) + C(0)
5 = B(1)(-1) ⇒B = -5
put x = 3
9 – 1 = A(0) + B(0) + C(3 – 1)(3 – 2)
8 = C(2)(1) ⇒ C = \(\frac{8}{2}\) = 4
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 48
= log (x – 1) -5 log (x – 2) + 4 log (x – 3) + C

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 3.
\(\int \frac{(\log x)^{n}}{x} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 49

Question 4.
\(\int \frac{1}{a^{2}-x^{2}} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 50
Finding A & B using parital fractions we get
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 51

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 5.
∫(2x2 – 6x + 4)3/2 (2x – 3) dx
Answer:
put 2x2 – 6x + 4 = t
(4x – 6) dx = dt
2(2x – 3) dx = dt
(2x – 3) dx = \(\frac{d t}{2}\)
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 52

Question 6.
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 53
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 54

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 7.
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 55
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 56
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 72
-x2e-x – 2xe-x + ∫e-x dx
= -x2e-x – 2xe-x – e-x + C

Question 8.
\(\int \frac{x^{2}}{(x-1)(x-2)(x-3)} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 57
By using partial fractions find the values A, B & C we get
A = \(\frac { 1 }{ 2 }\) , B = -4 , C = \(\frac { 9 }{ 2 }\)
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 58

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 9.
\(\int \frac{1}{x(x-2)(x+3)} d x\)
Answer:
By using partial fractions find the values of A, B & C we get
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 59

Question 10.
\(\int \frac{x^{3}}{x^{2}+7 x+12} d x\)
Answer:
This is an improper fraction
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 60
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 61

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 11.
\(\int \frac{3 x+2}{(x-1)(2 x+3)} d x\)
Answer:
= \(\int \frac{A}{x-1}+\frac{B}{2 x+3} d x\)
Resolving by partial fractions we get
A = 1, B = 1
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 62

Question 12.
\(\int \frac{2 x+3}{(x+1)^{2}(x-3)} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 63
By Resolving partial fractions
we get A = \(\frac { -9 }{ 16 }\), B = \(\frac { -1 }{ 4 }\), C = \(\frac { 9 }{ 16 }\)
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 64

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 13.
\(\int \frac{2 x}{\left(x^{2}+4 x+4\right)(x-1)} d x\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 65
By Resolving in to partial fractions find the values of A, B & C we get
A = \(\frac { -2 }{ 9 }\), B = \(\frac { 4 }{ 3 }\), C = \(\frac { 2 }{ 9 }\)

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 66

Question 14.
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 67
Answer:
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 68
Resolving partial fractions find the values of A &B we get A = -1, B = 1
Put log x = t
\(\frac{1}{x}\) dx = dt
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 69

2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Question 15.
\(\int \frac{e^{x}}{4 e^{2 x}-4 e^{x}-3} d x\)
Answer:
put ex = t ∴ex dx = dt
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 70
Find the values of A & B using partial fractions we get A = -1/4 , B = 1/4
2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals 71

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