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Integrals

Multiple Choice Questions

151.

$\int \cos \{2 \tan^{- 1} (\sqrt{\frac{1 - x}{1 + x}})\} dx$ is equal to

$\frac{1}{8} (x^{2} - 1) + C$
$\frac{x^{2}}{4} + C$
$\frac{x}{2} + C$
$\frac{x^{2}}{2} + C$

152.

$\int_{\frac{π}{6}}^{\frac{π}{3}} \frac{1}{\tan^{3} (x)} dx$ is

$\frac{π}{12}$
$\frac{π}{4}$
$\frac{π}{3}$
$\frac{π}{6}$

153.

$\int_{- 1}^{1} \frac{17 x^{5} - x^{4} + 29 x^{3} - 31 x + 1}{x^{2} + 1} dx$ is

154.

If I_n = $\int_{0}^{\frac{π}{4}} \tan^{n} (x) dx$ , then $\frac{1}{I_{3} + I_{5}}$ is

155.

If $\int_{0}^{\frac{π}{2}} \sin^{6} (x) dx = \frac{5 π}{32}$ , then the value of $\int_{- π}^{π} (\sin^{6} (x) + \cos^{6} (x)) dx$ is

$\frac{5 π}{8}$
$\frac{5 π}{16}$
$\frac{5 π}{2}$
$\frac{5 π}{4}$

156.

$\int \frac{\sec (x) \csc (x)}{2 \cot (x) - \sec (x) \csc (x)} dx$ is equal to

$\log |\sec (x) + \tan (x)|$ + c
$\log |\sec (x) + \csc (x)|$ + c
$\frac{1}{2} \log |\sec (2 x) + \tan (2 x)|$ + c
$\log |\sec (2 x) + \csc (2 x)|$ + c

157.
If $\int_{0}^{a} (2 a - x) dx = μ and \int_{0}^{a} f (x) dx = λ$ , then $\int_{0}^{2 a} f (x) dx$ equals

$2 λ - μ$

$λ + μ$

$μ - λ$

$λ - 2 μ$

$λ + μ$

$Given, \int_{0}^{a} f (2 a - x) dx = μ and \int_{0}^{a} f (x) dx = λ Now, using properties of definite integral \int_{0}^{2 a} f (x) dx = \int_{0}^{a} f (x) dx + \int_{0}^{a} f (2 a - x) dx \Rightarrow \int_{0}^{2 a} f (x) dx = λ + μ$

158.

$\int \tan (\sin^{- 1} (x)) dx$ is equal to

$\frac{1}{\sqrt{1 - x^{2}}} + c$
$- \sqrt{1 - x^{2}} + c$
$\frac{- x}{\sqrt{1 - x^{2}}} + c$
$\frac{x}{\sqrt{1 - x^{2}}} + c$

159.

$\int {(\sin (x) - \cos (x))}^{4} (\sin (x) + \cos (x)) dx$ is equal to

$\frac{\sin (x) - \cos (x)}{5} + c$
$\frac{{(\sin (x) - \cos (x))}^{5}}{5}$ + c
$\frac{{(\sin (x) - \cos (x))}^{4}}{4}$ + c
$\frac{{(\sin (x) + \cos (x))}^{5}}{5}$ + c

160.

$\int e^{\sin (θ)} [\log (\sin (θ)) + \csc^{2} (θ)] \cos (θ) dθ$ is equal to

$\int e^{\sin (θ)} [\log (\sin (θ)) + \csc^{2} (θ)] + c$
$e^{\sin (θ)} [\log (\sin (θ)) + \csc (θ)] + c$
$e^{\sin (θ)} [\log (\sin (θ)) - \csc (θ)]$
$e^{\sin (θ)} [\log (\sin (θ)) - \csc^{2} (θ)]$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

157. If ∫0a2a - xdx = μ and ∫0afxdx = λ, then ∫02afxdx equals 2λ - μ λ + μ μ - λ λ - 2μ

157.
If $\int_{0}^{a} (2 a - x) dx = μ and \int_{0}^{a} f (x) dx = λ$ , then $\int_{0}^{2 a} f (x) dx$ equals

$2 λ - μ$

$λ + μ$

$μ - λ$

$λ - 2 μ$