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451.
$\int_{0}^{π} \frac{xdx}{1 + \sin (x)}$ is equal to :

$- π$

$\frac{π}{2}$

$π$

None of these

$π$

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

452.

$\int \frac{dx}{x (x^{5} + 1)}$ is equal to :

$\frac{1}{5} \log (x^{5}) (x^{5} + 1) + c$
$\frac{1}{5} \log (\frac{x^{5} + 1}{x^{5}}) + c$
$\frac{1}{5} \log (\frac{x^{5}}{x^{5} + 1}) + c$
None of these

453.

$\int \frac{x + \sin (x)}{1 + \cos (x)} dx$ is equal to

$xtan (\frac{x}{2}) + c$
${xsec}^{2} (\frac{x}{2}) + c$
$\log (\cos (\frac{x}{2})) + c$
None of these

454.

$\int_{0}^{\frac{π}{8}} \cos^{3} (4 θ) dθ$ is equal to

$\frac{5}{3}$
$\frac{5}{4}$
$\frac{1}{3}$
$\frac{1}{6}$

455.

$\int_{0}^{\frac{π}{2}} \frac{\cos (x) - \sin (x)}{1 + \cos (x) \sin (x)} dx$ is equal to

0
$\frac{π}{2}$
$\frac{π}{4}$
$\frac{π}{6}$

456.

$\int \frac{dx}{x (x^{7} + 1)}$ is equal to

$\log (\frac{x^{7}}{x^{7} + 1}) + c$
$\frac{1}{7} \log (\frac{x^{7}}{x^{7} + 1}) + c$
$\log (\frac{x^{7} + 1}{x^{7}}) + c$
$\frac{1}{7} \log (\frac{x^{7} + 1}{x^{7}}) + c$

457.

$\int_{- 1}^{1} |1 - x| dx$ is equal to

458.

$\int \sqrt{x} e^{\sqrt{x}} dx$ is equal to

$2 \sqrt{x} - e^{\sqrt{x}} - 4 \sqrt{{xe}^{\sqrt{x}}} + c$
$(2 \sqrt{x} - 4 \sqrt{x} + 4) e^{\sqrt{x}} + c$
$(2 \sqrt{x} + 4 \sqrt{x} + 4) e^{\sqrt{x}} + c$
$(1 - 4 \sqrt{x}) e^{\sqrt{x}} + c$

459.

$\int \frac{dx}{x^{2} + 2 x + 2}$ is equal to

$\sin^{- 1} (x + 1) + c$
$\sinh^{- 1} (x + 1) + c$
$\tanh^{- 1} (x + 1) + c$
$\tan^{- 1} (x + 1) + c$

460.

$\int_{0}^{2 π} (\sin (x) + |\sin (x)|) dx$ is equal to

Previous Year Papers

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451.
$\int_{0}^{π} \frac{xdx}{1 + \sin (x)}$ is equal to :

$- π$

$\frac{π}{2}$

$π$

None of these

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

451. ∫0πxdx1 + sinx is equal to : - π π2 π None of these

451.
$\int_{0}^{π} \frac{xdx}{1 + \sin (x)}$ is equal to :

$- π$

$\frac{π}{2}$

$π$

None of these