2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise

Students can Download Maths Chapter 7 Integrals Miscellaneous Exercise Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise

Question 1.
\(\frac{1}{x-x^{3}}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 1

Question 2.
\(\frac{1}{\sqrt{x+a}+\sqrt{x+b}}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 2

KSEEB Solutions

Question 3.
\(\frac{1}{x \sqrt{a x-x^{2}}}\left[\text { Hint: Put } x=\frac{a}{t}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 3

Question 4.
\(\frac{1}{x^{2}\left(x^{4}+1\right)^{\frac{3}{4}}}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 4

Question 5.
\(\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}\)
\(\left[\text { Hint : } \frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}=\frac{1}{\frac{1}{x^{3}}\left(1+x^{\frac{1}{6}}\right)}, \text { put } x=t^{6}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 5

KSEEB Solutions

Question 6.
\(\frac{5 x}{(x+1)\left(x^{2}+9\right)}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 6
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 7

Question 7.
\(\frac{\sin x}{\sin (x-a)} d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 9

Question 8.
\(\frac{e^{5 \log x}-e^{4 \log x}}{e^{3 \log x}-e^{2 \log x}} d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 10

Question 9.
\(\frac{\cos x}{\sqrt{4-\sin ^{2} x}}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 11

Question 10.
\(\frac{\sin ^{8}-\cos ^{8} x}{1-2 \sin ^{2} x \cos ^{2} x}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 12

Question 11.
\(\frac{1}{\cos (x+a) \cos (x+b)}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 13
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 14

Question 12.
\(\frac{x^{3}}{\sqrt{1-x^{8}}}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 15

KSEEB Solutions

Question 13.
\(\frac{\mathbf{e}^{\mathbf{x}}}{\left(\mathbf{1}+\mathbf{e}^{\mathbf{x}}\right)\left(2+\mathbf{e}^{\mathbf{x}}\right)}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 16

Question 14.
\(\frac{1}{\left(x^{2}+1\right)\left(x^{2}+4\right)}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 17
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 18

Question 15.
cos3x elog sinx
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 19

Question 16.
e3 log x (x4+1)1
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 20

Question 17.
f'(ax+b)[f(ax+b)n
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 21

KSEEB Solutions

Question 18.
\(\frac{1}{\sqrt{\sin ^{3} x \sin (x+\alpha)}}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 22
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 23

Question 19.
\(\frac{\sin ^{-1} \sqrt{x}-\cos ^{-1} \sqrt{x}}{\sin ^{-1} \sqrt{x}+\cos ^{-1} \sqrt{x}}, x \in[0,1]\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 24
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 25

Question 20.
\(\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 26
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 27

KSEEB Solutions

Question 21.
\(\frac{2+\sin 2 x}{1+\cos 2 x} e^{x}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 28

Question 22.
\(\frac{x^{2}+x+1}{(x+1)^{2}(x+2)}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 29

Question 23.
\(\tan ^{-1} \sqrt{\frac{1-x}{1+x}}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 30
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 31

KSEEB Solutions

Question 24.
\(\frac{\sqrt{x^{2}+1}\left[\log \left(x^{2}+1\right)-2 \log x\right]}{x^{4}}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 32

Evaluate the definite integrals in Exercises 25 to 33

Question 25.
\(\int_{\frac{\pi}{2}}^{\pi} \mathrm{e}^{x}\left(\frac{1-\sin x}{1+\cos x}\right) d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 33

Question 26.
\(\int_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos ^{4} x+\sin ^{4} x} d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 34
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 35

KSEEB Solutions

Question 27.
\(\int_{0}^{\frac{\pi}{2}} \frac{\cos ^{2} x d x}{\cos ^{2} x+4 \sin ^{2} x}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 36
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 37
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 38
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 39
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 40

Question 28.
\(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x+\cos x}{\sqrt{\sin 2 x}} d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 41
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 42
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 43

Question 29.
\(\int_{0}^{1} \frac{d x}{\sqrt{1+x}-\sqrt{x}}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 44

KSEEB Solutions

Question 30.
\(\int_{0}^{\frac{\pi}{4}} \frac{\sin x+\cos x}{9+16 \sin 2 x} d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 45
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 46

Question 31.
\(\int_{0}^{\frac{\pi}{2}} \sin 2 x \tan ^{-1}(\sin x) d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 47

Question 32.
\(\int_{0}^{\pi} \frac{x \tan x}{\sec x+\tan x} d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 48
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 49

Question 33.
\(\int_{1}^{4}[|x-1|+|x-2|+|x-3|] d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 50
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 51

Prove the following (Exercise 34 to 39)

Question 34.
\(\int_{1}^{3} \frac{d x}{x^{2}(x+1)}=\frac{2}{3}+\log \frac{2}{3}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 52

Question 35.
\(\int_{0}^{1} x e^{x} d x=1\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 53

KSEEB Solutions

Question 36.
\(\int_{-1}^{1} x^{17} \cos ^{4} x d x=0\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 54

Question 37.
\(\int_{0}^{\frac{\pi}{2}} \sin ^{3} x d x=\frac{2}{3}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 55

Question 38.
\(\int_{0}^{\frac{\pi}{4}} 2 \tan ^{3} x d x=1-\log 2\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 56

Question 39.
\(\int_{0}^{1} \sin ^{-1} x d x=\frac{\pi}{2}-1\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 57

Question 40.
Evaluate \(\int_{0}^{1} e^{2-3 x} d x\) as a a= limit of a sum.
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 58

KSEEB Solutions

Choose the correct answers in Exercises 41 and 44.

Question 41.
\(\int \frac{d x}{e^{x}+e^{-x}} \text { is equal to }\)
(A) tan-1 (ex) + C
(B) tan1 (e x) + C
(C) log (ex – e -x) + C
(D) log (ex + e-x) + C
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 59

Question 42.
\(\int \frac{\cos 2 x}{(\sin x+\cos x)^{2}} d x \text { is equal to }\)
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 60
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 61

Question 43.
If f (a+b-x) = f (x), then \(\int_{a}^{b} x f(x) d x \)is equal to
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 62
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 63

Question 44.
The value of \(\int_{0}^{1} \tan ^{-1}\left(\frac{2 x-1}{1+x-x^{2}}\right) d x \text { is }\)
(A) 1
(B) 0
(C) -1
(D) \(\frac{\pi}{4}\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 64

2nd PUC Maths Integrals Miscellaneous Exercise Extra Questions and Answers

Question 1.
\(\int \frac{x}{1+x^{4}} d x=\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 65

Question 2.
\(\int \frac{x^{3}}{1+x^{4}} d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 66

Question 3.
\(\int \frac{x^{2}}{1+x^{4}} d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 67
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 68

KSEEB Solutions

Question 4.
\(\int \frac{1}{1+x^{4}} d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 69

Question 5.
\(\int \sqrt{\cot x} d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 70
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 71
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 72

Question 6.
\(\int \sqrt{\tan x} d x\) (CBSE 2009)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 73
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 74
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 75

Question 7.
\(\int \frac{x^{2}+1}{1+x^{4}} d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 76

Question 8.
\(\text { If } \int \frac{1}{f(x)} d x=\log (f(x))^{2}+C, \text { find } f(x)\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 77
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 78

Question 9.
\(\int e^{\tan ^{-1} x}\left(1+\frac{x}{1+x^{2}}\right) d x\)(CBSE 2006)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 79

Question 10.
\(\int \sin ^{-1} \sqrt{\frac{x}{1+x}} d x\) (CBSE 2010)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 80

KSEEB Solutions

Question 11.
∫|x| dx
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 81

Question 12.
\(\int\left(\log x+\frac{1}{(\log x)^{2}}\right) d x\)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 82

Question 13.
∫eax (af(x) + f'(x))dx = f (x). eax
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 83

Question 14.
∫e 3-x(cosx – 3sinx) dx   (Kerala CET)
Answer:
∫ e-3x (-3 sin x + cos x)
f(x) = sinx,f'(x) =cosx
∫ e-3x (-3 sin x + cos x) = e-3x sin x + C

Question 15.
∫ sec2 (7-4x) dx (CBSE – 2010)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 84

Question 16.
\(\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sin 5 x \cdot d x\) (CBSE – 2010)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 85

Question 17.
\(\int \frac{x+2}{\sqrt{(x-2)(x-3)}} d x\) (CBSE – 2010)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 86
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 87

KSEEB Solutions

Question 18.
\(\int \frac{5 x+3}{\sqrt{x^{2}+4 x+10}} d x\) (CBSE – 2010)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 88

Question 19.
∫(ax + b)3 dx (CBSE – 2011)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 89

Question 20.
\(\int_{0}^{\frac{\pi}{2}} \frac{x+\sin x}{1+\cos x} d x\) (CBSE – 2011)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 90

Question 21.
\(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{d x}{1+\sqrt{\tan x}} d x\)(CBSE – 2011)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 91

Question 22.
\(\int \frac{(\log x)^{2}}{x} d x\) (CBSE – 2011)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 92

Question 23.
\(\int_{0}^{\frac{\pi}{4}} \log (1+\tan x) d x\) (CBSE – 2011)
Answer:
\(\frac{\pi}{8} \log 2\)
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 93
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 94
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 95

Question 24.
\(\int_{-6}^{0}|x+3| d x\) (CBSE – 2011)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 96
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 97

Question 25.
\(\int \frac{6 x+7}{(x-5)(x-4)} d x\) (CBSE – 2011)
Answer:
2nd PUC Maths Question Bank Chapter 7 Integrals Miscellaneous Exercise 98

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