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:

Question 2.
\(\frac{1}{\sqrt{x+a}+\sqrt{x+b}}\)
Answer:

Question 3.
\(\frac{1}{x \sqrt{a x-x^{2}}}\left[\text { Hint: Put } x=\frac{a}{t}\right]\)
Answer:

Question 4.
\(\frac{1}{x^{2}\left(x^{4}+1\right)^{\frac{3}{4}}}\)
Answer:

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:

Question 6.
\(\frac{5 x}{(x+1)\left(x^{2}+9\right)}\)
Answer:

Question 7.
\(\frac{\sin x}{\sin (x-a)} d x\)
Answer:

Question 8.
\(\frac{e^{5 \log x}-e^{4 \log x}}{e^{3 \log x}-e^{2 \log x}} d x\)
Answer:

Question 9.
\(\frac{\cos x}{\sqrt{4-\sin ^{2} x}}\)
Answer:

Question 10.
\(\frac{\sin ^{8}-\cos ^{8} x}{1-2 \sin ^{2} x \cos ^{2} x}\)
Answer:

Question 11.
\(\frac{1}{\cos (x+a) \cos (x+b)}\)
Answer:

Question 12.
\(\frac{x^{3}}{\sqrt{1-x^{8}}}\)
Answer:

Question 13.
\(\frac{\mathbf{e}^{\mathbf{x}}}{\left(\mathbf{1}+\mathbf{e}^{\mathbf{x}}\right)\left(2+\mathbf{e}^{\mathbf{x}}\right)}\)
Answer:

Question 14.
\(\frac{1}{\left(x^{2}+1\right)\left(x^{2}+4\right)}\)
Answer:

Question 15.
cos³x e^{log sinx}
Answer:

Question 16.
e^{3 log x} (x⁴+1)¹
Answer:

Question 17.
f'(ax+b)[f(ax+b)ⁿ
Answer:

Question 18.
\(\frac{1}{\sqrt{\sin ^{3} x \sin (x+\alpha)}}\)
Answer:

Question 19.
\(\frac{\sin ^{-1} \sqrt{x}-\cos ^{-1} \sqrt{x}}{\sin ^{-1} \sqrt{x}+\cos ^{-1} \sqrt{x}}, x \in[0,1]\)
Answer:

Question 20.
\(\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}\)
Answer:

Question 21.
\(\frac{2+\sin 2 x}{1+\cos 2 x} e^{x}\)
Answer:

Question 22.
\(\frac{x^{2}+x+1}{(x+1)^{2}(x+2)}\)
Answer:

Question 23.
\(\tan ^{-1} \sqrt{\frac{1-x}{1+x}}\)
Answer:

Question 24.
\(\frac{\sqrt{x^{2}+1}\left[\log \left(x^{2}+1\right)-2 \log x\right]}{x^{4}}\)
Answer:

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:

Question 26.
\(\int_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos ^{4} x+\sin ^{4} x} d x\)
Answer:

Question 27.
\(\int_{0}^{\frac{\pi}{2}} \frac{\cos ^{2} x d x}{\cos ^{2} x+4 \sin ^{2} x}\)
Answer:

Question 28.
\(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x+\cos x}{\sqrt{\sin 2 x}} d x\)
Answer:

Question 29.
\(\int_{0}^{1} \frac{d x}{\sqrt{1+x}-\sqrt{x}}\)
Answer:

Question 30.
\(\int_{0}^{\frac{\pi}{4}} \frac{\sin x+\cos x}{9+16 \sin 2 x} d x\)
Answer:

Question 31.
\(\int_{0}^{\frac{\pi}{2}} \sin 2 x \tan ^{-1}(\sin x) d x\)
Answer:

Question 32.
\(\int_{0}^{\pi} \frac{x \tan x}{\sec x+\tan x} d x\)
Answer:

Question 33.
\(\int_{1}^{4}[|x-1|+|x-2|+|x-3|] d x\)
Answer:

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:

Question 35.
\(\int_{0}^{1} x e^{x} d x=1\)
Answer:

Question 36.
\(\int_{-1}^{1} x^{17} \cos ^{4} x d x=0\)
Answer:

Question 37.
\(\int_{0}^{\frac{\pi}{2}} \sin ^{3} x d x=\frac{2}{3}\)
Answer:

Question 38.
\(\int_{0}^{\frac{\pi}{4}} 2 \tan ^{3} x d x=1-\log 2\)
Answer:

Question 39.
\(\int_{0}^{1} \sin ^{-1} x d x=\frac{\pi}{2}-1\)
Answer:

Question 40.
Evaluate \(\int_{0}^{1} e^{2-3 x} d x\) as a a= limit of a sum.
Answer:

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 (e^x) + C
(B) tan¹ (e ^x) + C
(C) log (e^x – e ^-x) + C
(D) log (e^x + e^-x) + C
Answer:

Question 42.
\(\int \frac{\cos 2 x}{(\sin x+\cos x)^{2}} d x \text { is equal to }\)

Answer:

Question 43.
If f (a+b-x) = f (x), then \(\int_{a}^{b} x f(x) d x \)is equal to

Answer:

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 Integrals Miscellaneous Exercise Extra Questions and Answers

Question 1.
\(\int \frac{x}{1+x^{4}} d x=\)
Answer:

Question 2.
\(\int \frac{x^{3}}{1+x^{4}} d x\)
Answer:

Question 3.
\(\int \frac{x^{2}}{1+x^{4}} d x\)
Answer:

Question 4.
\(\int \frac{1}{1+x^{4}} d x\)
Answer:

Question 5.
\(\int \sqrt{\cot x} d x\)
Answer:

Question 6.
\(\int \sqrt{\tan x} d x\) (CBSE 2009)
Answer:

Question 7.
\(\int \frac{x^{2}+1}{1+x^{4}} d x\)
Answer:

Question 8.
\(\text { If } \int \frac{1}{f(x)} d x=\log (f(x))^{2}+C, \text { find } f(x)\)
Answer:

Question 9.
\(\int e^{\tan ^{-1} x}\left(1+\frac{x}{1+x^{2}}\right) d x\)(CBSE 2006)
Answer:

Question 10.
\(\int \sin ^{-1} \sqrt{\frac{x}{1+x}} d x\) (CBSE 2010)
Answer:

Question 11.
∫|x| dx
Answer:

Question 12.
\(\int\left(\log x+\frac{1}{(\log x)^{2}}\right) d x\)
Answer:

Question 13.
∫e^ax (af(x) + f'(x))dx = f (x). e^ax
Answer:

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:

Question 16.
\(\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sin 5 x \cdot d x\) (CBSE – 2010)
Answer:

Question 17.
\(\int \frac{x+2}{\sqrt{(x-2)(x-3)}} d x\) (CBSE – 2010)
Answer:

Question 18.
\(\int \frac{5 x+3}{\sqrt{x^{2}+4 x+10}} d x\) (CBSE – 2010)
Answer:

Question 19.
∫(ax + b)³ dx (CBSE – 2011)
Answer:

Question 20.
\(\int_{0}^{\frac{\pi}{2}} \frac{x+\sin x}{1+\cos x} d x\) (CBSE – 2011)
Answer:

Question 21.
\(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{d x}{1+\sqrt{\tan x}} d x\)(CBSE – 2011)
Answer:

Question 22.
\(\int \frac{(\log x)^{2}}{x} d x\) (CBSE – 2011)
Answer:

Question 23.
\(\int_{0}^{\frac{\pi}{4}} \log (1+\tan x) d x\) (CBSE – 2011)
Answer:
\(\frac{\pi}{8} \log 2\)

Question 24.
\(\int_{-6}^{0}|x+3| d x\) (CBSE – 2011)
Answer:

Question 25.
\(\int \frac{6 x+7}{(x-5)(x-4)} d x\) (CBSE – 2011)
Answer: