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Integrals

Multiple Choice Questions

631.

If I₁ = $\int \sin^{- 1} (x) dx$ and I₂ = $\int \sin^{- 1} (\sqrt{1 - x^{2}}) dx$ , then

I₁ = I₂
$I_{2} = \frac{π}{2} I_{1}$
$I_{1} + I_{2} = \frac{π}{2} x$
$I_{1} + I_{2} = \frac{π}{2}$

632.

$\int \frac{(\sin (θ) + \cos (θ))}{\sqrt{\sin (2 θ)}} dθ$ is equal to

$\log |\cos (θ) - \sin (θ) + \sqrt{\sin (2 θ)}| + c$
$\log |\sin (θ) - \cos (θ) + \sqrt{\sin (2 θ)}| + c$
$\sin^{- 1} (\sin (θ) - \cos (θ)) + c$
$\sin^{- 1} (\sin (θ) + \cos (θ)) + c$

633.

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

$\frac{π}{12}$
$\frac{π}{2}$
$\frac{π}{6}$
$\frac{π}{4}$

634.

If f is a continuous function, then

$\int_{- 2}^{2} f (x) dx = \int_{0}^{2} [f (x) - f (- x)] dx$
$\int_{- 3}^{5} 2 f (x) dx = \int_{- 6}^{10} f (x - 1) dx$
$\int_{- 3}^{5} f (x) dx = \int_{- 4}^{4} f (x - 1) dx$
$\int_{- 3}^{5} f (x) dx = \int_{- 2}^{6} f (x - 1) dx$

635.

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

$xtan (\frac{x}{2}) + c$
$\log (1 + \cos (x)) + c$
$\cot (\frac{x}{2}) + c$
$\log (x + \sin (x)) + c$

636.

$\int \cos^{3} (x) . e^{\log (\sin (x))} dx$ is equal to

$- \frac{\sin^{4} (x)}{4} + c$
$- \frac{\cos^{4} (x)}{4} + c$
$\frac{e^{\sin (x)}}{4} + c$
None of the above

637.
The value of $\int_{0}^{\frac{π}{2}} \frac{\cos (3 x) + 1}{2 \cos (x) - 1} dx$ is

2

1

$\frac{1}{2}$

0

$Let I = \int_{0}^{\frac{π}{2}} \frac{\cos (3 x) + 1}{2 \cos (x) - 1} dx = \int_{0}^{\frac{π}{2}} \frac{\cos (3 x) - \cos (\frac{3 π}{3})}{2 (\cos (x) - \cos (\frac{π}{3}))} dx = \int_{0}^{\frac{π}{2}} \frac{(4 \cos^{3} (x) - 3 \cos (x)) - (4 \cos^{3} (\frac{π}{3}) - 3 \cos (\frac{π}{3}))}{2 (\cos (x) - \cos (\frac{π}{3}))} dx = 2 \int_{0}^{\frac{π}{2}} (\frac{\cos^{3} (x) - \cos^{3} (\frac{π}{3})}{\cos (x) - \cos (\frac{π}{3})}) dx - \frac{3}{2} \int_{0}^{\frac{π}{2}} (\frac{\cos (x) - \cos (\frac{π}{3})}{\cos (x) - \cos (\frac{π}{3})}) dx = 2 \int_{0}^{\frac{π}{2}} (\cos^{2} (x) + \cos^{2} (\frac{π}{3}) + \cos (x) \cos (\frac{π}{3})) - \frac{3}{2} \int_{0}^{\frac{π}{2}} 1 dx = \int_{0}^{\frac{π}{2}} (1 + \cos (2 x) + \frac{1}{2} + \cos (x)) dx - \frac{3 π}{4} = \frac{3 π}{4} + 1 - \frac{3 π}{4} = 1$