Bihar Board 12th Maths Objective Answers Chapter 7 Integrals

Bihar Board 12th Maths Objective Questions and Answers

Bihar Board 12th Maths Objective Answers Chapter 7 Integrals

Question 1.
Evaluate: ∫(2tan x – 3cot x)² dx
(a) -4tan x – 9cot x – 25x + C
(b) 4tan x – 9cot x – 25x + C
(c) -4tan x + 9 cot x + 25x + C
(d) 4tan x + 9cot x + 25x + C
Answer:
(b) 4tan x – 9cot x – 25x + C

Question 2.

Answer:
(a) \(\frac{a^{x}}{\log a}+\frac{x^{a+1}}{a+1}+a^{a} x+C\)

Question 3.

Answer:
(a) \(-\frac{1}{4} \tan (7-4 x)+C\)

Question 4.

Answer:
(a) \(\frac{1}{(\log 2)^{3}} 2^{2^{2^{x}}}+C\)

Question 5.

Answer:
(b) \(-\frac{\cos ^{4} x}{4}+C\)

Question 6.

Answer:
(a) \(\frac{-3}{\sqrt[3]{\sin x}}+C\)

Question 7.
Evaluate: ∫tan(x – θ) tan(x + θ) tan2x dx
(a) \(\frac { 1 }{ 2 }\) log|cos2x| – log|cos(x – θ)| + log|cos(x + θ)| + C
(b) –\(\frac { 1 }{ 2 }\) log|cos2x| + log|cos(x – θ)| + log|cos (x + θ)| + C
(c) –\(\frac { 1 }{ 2 }\) log|cos2x| – log|cos(x – θ)| – log|cos(x + θ)| + C
(d) None of these
Answer:
(b) –\(\frac { 1 }{ 2 }\) log|cos2x| + log|cos(x – θ)| + log|cos (x + θ)| + C

Question 8.

Answer:
(b) \(-\frac{2}{\sqrt{\tan x}}+\frac{2}{3}(\tan x)^{3 / 2}+C\)

Question 9.

Answer:
(b) \(-\frac{3}{5} \tan ^{-5 / 3} x+3 \tan ^{1 / 3} x+C\)

Question 10.

Answer:
(c) \(\frac{1}{4} \log \left|x^{4}-9\right|-\frac{1}{12}\left|\frac{x^{2}-3}{x^{2}+3}\right|+C\)

Question 11.

Answer:
(b) \(-\log \left|e^{-x}+\sqrt{e^{-2 x}-1}\right|+C\)

Question 12.

Answer:
(b) \(\frac{1}{n} \log \left|\frac{x^{n}}{x^{n}+1}\right|+C\)

Question 13.

Answer:
(c) \(\frac{1}{6} \tan ^{-1}\left(\frac{2 \tan x}{3}\right)+C\)

Question 14.

Answer:
(c) \(2 \sqrt{x}-3(\sqrt[3]{x})+6(\sqrt[6]{x})-6 \log (\sqrt[6]{x}+1)+C\)

Question 15.

Answer:
(b) \(\begin{array}{l}
-\frac{1}{3} \log |1+\tan \theta|+\frac{1}{6} \log \left|\tan ^{2} \theta-\tan \theta+1\right| \\ \quad+\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{2 \tan \theta-1}{\sqrt{3}}\right)+C
\end{array}\)

Question 16.

Answer:
(a) \(\log |\sec x+\tan x|-2 \tan (x / 2)+C\)

Question 17.

Answer:
(c) \(\tan \frac{x}{2}\)

Question 18.

Answer:
(a) \(\frac{e^{x} \cos x}{1+\sin x}+C\)

Question 19.

Answer:
(b) \(e^{x}\left(\frac{x+2}{x+4}\right)+C\)

Question 20.

Answer:
(b) \(\begin{aligned} \frac{1}{2}(x+1) \sqrt{x^{2}+2 x+5} & \\ +2 \log |(x+1)+\sqrt{x^{2}+2 x+5}|+C & \end{aligned}\)

Question 21.
\(\int_{1}^{2} x^{2} d x\)
(a) 1
(b) \(\frac{7}{3}\)
(c) \(\frac{1}{3}\)
(d) 0
Answer:
(b) \(\frac{7}{3}\)

Question 22.
\(\int_{0}^{2}\left(x^{2}+3\right) d x\)
(a) \(\frac{25}{3}\)
(b) \(\frac{26}{3}\)
(c) \(\frac{24}{3}\)
(d) None of these
Answer:
(b) \(\frac{26}{3}\)

Question 23.
Evaluate: \(\int_{0}^{\pi / 4} \sqrt{1-\sin 2 x} d x\)
(a) √2 – 1
(b) √2 + 1
(c) √2
(d) None of these
Answer:
(a) √2 – 1

Question 24.
Evaluate: \(\int_{0}^{2 \pi} \sin \left(\frac{\pi}{4}+\frac{x}{2}\right) d x\)
(a) -2√2
(b) -2
(c) √2
(d) 2√2
Answer:
(d) 2√2

Question 25.
Evaluate: \(\int_{1}^{2} \frac{d x}{x^{2}}\)
(a) \(\frac{1}{2}\)
(b) 1
(c) 2
(d) -1
Answer:
(a) \(\frac{1}{2}\)

Question 26.
Evaluate: \(\int_{0}^{1} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x\)
(a) \(\frac{\pi}{2}\) – log2
(b) π
(c) \(\frac{\pi}{4}\)
(d) \(\frac{\pi}{2}\) – log2
Answer:
(a) \(\frac{\pi}{2}\) – log2

Question 27.

Answer:
(a) \(\log \left(\frac{4}{3}\right)\)

Question 28.

Answer:
(c) \(\frac{4-\pi}{4 \sqrt{2}}\)

Question 29.

Answer:
(c) \(\frac{2}{\sqrt{5}} \tan ^{-1}\left(\frac{1}{\sqrt{5}}\right)\)

Question 30.

Answer:
(c) \(\frac{1}{\sqrt{5}} \log \left(\frac{3+\sqrt{5}}{2}\right)\)

Question 31.

Answer:
(a) \(\frac{\pi}{2}\)

Question 32.

Answer:
(a) \(\frac{8}{21}\)

Question 33.

Answer:
(a) 2 – √2

Question 34.

Answer:
(b) \(\frac{1}{2}+\frac{1}{\pi+2}-A\)

Question 35.

Answer:
(d) \(\frac{\pi}{2}\)

Question 36.

Answer:
(c) π²

Question 37.

Answer:
(c) \(\operatorname{cosec}(b-a) \log \left|\frac{\sin (x-b)}{\sin (x-a)}\right|+C\)

Question 38.

Answer:
(a) \(\frac{e^{x}}{1+x^{2}}+C\)

Question 39.

Answer:
(d) \(x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\log |1+x|+C\)

Question 40.

Answer:
(d) a = \(\frac{1}{3}\), b = -1

Question 41.

Answer:
(a) 1