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Integrals

Multiple Choice Questions

131.

The value of integral $\int_{0}^{1} \sqrt{\frac{1 - x}{1 + x}} d x$ is

$\frac{π}{2} + 1$
$\frac{π}{2} - 1$
- 1
1

132.

Evaluate $\int \frac{x^{2} + 4}{x^{4} + 16} dx$ .

$\frac{1}{2 \sqrt{2}} \tan^{- 1} (\frac{x^{2} - 4}{2 x \sqrt{2}}) + C$
$\frac{1}{2 \sqrt{2}} \tan^{- 1} (\frac{x^{2} - 4}{2 \sqrt{2}}) + C$
$\frac{1}{2 \sqrt{2}} \tan^{- 1} (\frac{x^{2} - 4}{x \sqrt{2}}) + C$
None of these

133.

Evaluate $\int_{\frac{π}{4}}^{\frac{3 π}{4}} \frac{1}{1 + \cos (x)} dx$

2
- 2
1/2
- 1/2

134.

If $\int f (x) dx = f (x), then \int {\{f (x)\}}^{2} dx$ is equal to

$\frac{1}{2} {\{f (x)\}}^{2}$
${\{f (x)\}}^{3}$
$\frac{{\{f (x)\}}^{3}}{3}$
${\{f (x)\}}^{2}$

135.

$\int \sin^{- 1} \{\frac{(2 x + 2)}{\sqrt{4 x^{2} + 8 x + 13}}\} dx$ is equal to

$(x + 1) \tan^{- 1} (\frac{2 x + 2}{3}) - \frac{3}{4} \log (\frac{4 x^{2} + 8 x + 13}{9}) + c$
$\frac{3}{2} \tan^{- 1} (\frac{2 x + 2}{3}) - \frac{3}{4} og (\frac{4 x^{2} + 8 x + 13}{9}) + c$
$(x + 1) \tan^{- 1} (\frac{2 x + 2}{3}) - \frac{3}{2} l og (4 x^{2} + 8 x + 13) + c$
$\frac{3}{2} (x + 1) \tan^{- 1} (\frac{2 x + 2}{3}) - \frac{3}{4} \log (4 x^{2} + 8 x + 13) + c$

136.
If $\int_{1}^{x} \frac{d t}{|t| \sqrt{t^{2} - 1}} = \frac{π}{6}$ , then x can be equal to

$\frac{2}{\sqrt{3}}$

$\sqrt{3}$

2

None of these

$\frac{2}{\sqrt{3}}$

$\int_{1}^{x} \frac{dπt}{|t| \sqrt{t^{2} - 1}} = \frac{π}{6} \Rightarrow {[\sec^{- 1} (t)]}_{1}^{x} = \frac{π}{6} \Rightarrow \sec^{- 1} (x) - \sec^{- 1} (1) = \frac{π}{6} \Rightarrow \sec^{- 1} (x) - 0 = \frac{π}{6} \Rightarrow x = \sec (\frac{π}{6}) \Rightarrow x = \frac{2}{\sqrt{3}}$

137.

$\int (\frac{2 - \sin (2 x)}{1 - \cos (2 x)}) e^{x} dx$ is equal to

$- e^{x} \cot (x) + c$
$e^{x} \cot (x) + c$
$2 e^{x} \cot (x) + c$
$- 2 e^{x} \cot (x) + c$

138.

$If I_{n} = \int \sin^{n} (x) dx, then {nI}_{n} - (n - 1) I_{n - 2}$ equals

$\sin^{n - 1} (x) xcos (x)$
$\cos^{n - 1} (x) \sin (x)$
$- \sin^{n - 1} (x) \cos (x)$
$- \cos^{n - 1} (x) \sin (x)$

139.

If I = $\int \frac{x^{5}}{\sqrt{1 + x^{3}}} dx$ , then I is equal to

$\frac{2}{9} {(1 + x^{3})}^{\frac{5}{2}} + \frac{2}{3} {(1 + x^{3})}^{\frac{3}{2}} + c$
$\log |\sqrt{x} + \sqrt{1 + x^{3}}| + c$
$\log |\sqrt{x} - \sqrt{1 + x^{3}}| + c$
$\frac{2}{9} {(1 + x^{3})}^{\frac{3}{2}} - \frac{2}{3} {(1 + x^{3})}^{\frac{1}{2}} + c$