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Integrals

Multiple Choice Questions

451.

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

$- π$
$\frac{π}{2}$
$π$
None of these

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

4

0

1

8

$We have, \int_{0}^{2 π} (\sin (x) + |\sin (x)|) dx = \int_{0}^{π} (\sin (x) + |\sin (x)|) dx + \int_{π}^{2 π} (\sin (x) + |\sin (x)|) dx [Using \int_{0}^{2 a} f (x) dx = \int_{0}^{a} f (x) dx + \int_{0}^{2 a} f (x) dx] = \int_{0}^{π} (\sin (x) + \sin (x)) dx + \int_{π}^{2 π} (\sin (x) - \sin (x)) dx = \int_{0}^{π} 2 \sin (x) dx + 0 = 2 \int_{0}^{π} 2 \sin (x) dx = 2 {[- \cos (x)]}_{0}^{π} = - 2 (\cos (π) - \cos (0)) = - 2 (- 1 - 1) = 4$

Previous Year Papers

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Multiple Choice Questions

460. ∫02πsinx + sinxdx is equal to 4 0 1 8

460.
$\int_{0}^{2 π} (\sin (x) + |\sin (x)|) dx$ is equal to

4

0

1

8