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Integrals

Multiple Choice Questions

531.

$\int_{0}^{\frac{π}{2}} \frac{1}{a^{2} . \sin^{2} (x) + b^{2} . \cos^{2} (x)} dx$

$\frac{π}{2 ab}$
$\frac{πb}{4 a}$
$\frac{πa}{2 b}$
$\frac{πa}{4 b}$

532.

$\int \sqrt{x^{2} + 2 x + 5} dx$ is equal to

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

533.

$\int \frac{(x + 3) e^{x}}{{(x + 4)}^{2}} dx$ is equal to

$\frac{e^{x}}{{(x + 4)}^{2}} + C$
$\frac{e^{x}}{(x + 3)} + C$
$\frac{1}{{(x + 4)}^{2}} + C$
$\frac{e^{x}}{(x + 4)} + C$

534.
$\int \frac{\cos (2 x) - \cos (2 θ)}{\cos (x) - \cos (θ)} dx$ is equal to

$2 (\sin (x) + xcos (θ))$

$2 (\sin (x) - xcos (θ))$

$2 (\sin (x) + 2 xcos (θ))$

$2 (\sin (x) - 2 xcos (θ))$

$2 (\sin (x) + xcos (θ))$

$Let I = \int \frac{\cos (2 x) - \cos (2 θ)}{\cos (x) - \cos (θ)} dx = \int \frac{(2 \cos^{2} (x) - 1) - (2 \cos^{2} (θ) - 1)}{\cos (x) - \cos (θ)} dx = 2 \int \frac{\cos^{2} (x) - \cos^{2} (θ)}{\cos (x) - \cos (θ)} dx = 2 \int \frac{(\cos (x) - \cos (θ)) (\cos (x) - \cos (θ))}{\cos (x) - \cos (θ)} dx = 2 \int (\cos (x) + \cos (θ)) = 2 (\sin (x) + xcos (θ)) + C$