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Integrals

Multiple Choice Questions

731.

The value of the integral $\int_{a}^{b} \frac{\sqrt{x} dx}{\sqrt{x} + \sqrt{a + b - x}}$ is :

$π$
$\frac{1}{2} (b - a)$
$π / 2$
b - a

732.

$\int_{0}^{\frac{π}{2}} \frac{\sqrt{\cot (x)}}{\sqrt{\cot (x)} + \sqrt{\tan (x)}} dx$ is equal to :

1
- 1
$\frac{π}{2}$
$\frac{π}{4}$

733.

$\int_{0}^{\frac{π}{2}} {xsin}^{2} (x) \cos^{2} (x) dx$ is equal to :

$\frac{π^{2}}{32}$
$\frac{π^{2}}{16}$
$\frac{π}{32}$
None of these

734.

$\int_{- \frac{π}{3}}^{\frac{π}{3}} \frac{xsin (x)}{\cos^{2} (x)} dx$ is :

$\frac{1}{3} (4 π + 1)$
$\frac{4 π}{3} - 2 \log (\tan (\frac{5 π}{12}))$
$\frac{4 π}{3} + \log (\tan (\frac{5 π}{12}))$
None of these

735.

$\int \frac{{(\tan^{- 1} (x))}^{3}}{(1 + x^{2})} dx$ is equa to :

$3 {(\tan^{- 1} (x))}^{2} + c$
$\frac{{(\tan^{- 1} (x))}^{4}}{4} + c$
${(\tan^{- 1} (x))}^{4} + c$
None of these

736.
$\int_{0}^{π} \frac{xdx}{1 + \sin (x)}$ is equal to :

$- π$

$\frac{π}{2}$

$π$

None of these

$π$

$Let I = \int_{0}^{π} \frac{xdx}{1 + \sin (x)} . . . (i) = \int_{0}^{π} \frac{(π - x) dx}{1 + \sin (π - x)} = \int_{0}^{π} \frac{(π - x) dx}{1 + \sin (x)} . . . (ii) On adding Eqs . (i) and (ii), 2 I = \int_{0}^{π} \frac{πdx}{1 + \sin (x)} \Rightarrow 2 I = π \int_{0}^{π} \frac{1 - \sin (x)}{1 - \sin^{2} (x)} dx \Rightarrow 2 I = π \int_{0}^{π} (\sec^{2} (x) - \sec (x) \tan (x)) dx \Rightarrow 2 I = π {[\tan (x) - \sec (x)]}_{0}^{π} \Rightarrow 2 I = π [0 - (- 1) - (0 - 1)] \Rightarrow 2 I = 2 π \Rightarrow I = π$