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Integrals

Multiple Choice Questions

291.

$\int e^{xlog (a)} e^{x} dx$ is equal to :

$\frac{a^{x}}{\log (ae)} + c$
$\frac{e^{x}}{1 + \log_{e} (a)} + c$
${(ae)}^{x} + c$
$\frac{{(ae)}^{x}}{\log_{e} (ae)} + c$

292.

$\int \sqrt{e^{x} - 1} dx$ is equal to :

$2 [\sqrt{e^{x} - 1} - \tan^{- 1} (\sqrt{e^{x} - 1})] + c$
$\sqrt{e^{x} - 1} - \tan^{- 1} (\sqrt{e^{x} - 1}) + c$
$\sqrt{e^{x} - 1} + \tan^{- 1} (\sqrt{e^{x} - 1}) + c$
$2 [\sqrt{e^{x} - 1} + \tan^{- 1} (\sqrt{e^{x} - 1})] + c$

293.

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

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

294.

$\int \cos^{\frac{- 3}{7}} (x) \sin^{\frac{- 11}{7}} (x) dx$ is equal to:

$\log |\sin^{\frac{4}{7}} (x)| + c$
$\frac{4}{7} \tan^{\frac{4}{7}} (x) + c$
$\frac{- 7}{4} \tan^{\frac{- 4}{7}} (x) + c$
$\log |\cos^{\frac{3}{7}} (x)| + c$

295.

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

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

296.
$\int_{\frac{π}{6}}^{\frac{π}{3}} \frac{dx}{1 + \sqrt{\tan (x)}}$ is equal to :

$\frac{π}{12}$

$\frac{π}{2}$

$\frac{3 π}{2}$

$2 π$

$\frac{π}{12}$

$Let \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{dx}{1 + \sqrt{\tan (x)}} dx \Rightarrow I = \int_{\frac{π}{6}}^{\frac{π}{3}} (\frac{\sqrt{\cos (x)}}{\sqrt{\sin (x)} + \sqrt{\cos (x)}}) dx . . . (i) = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{\sqrt{\cos (\frac{π}{2} - x)}}{\sqrt{\sin (\frac{π}{2} - x) \cos (\frac{π}{2} - x)}} dx = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{\sqrt{\sin (x)}}{\sqrt{\cos (x)} + \sqrt{\sin (x)}} dx . . . (ii) 2 I = \int_{\frac{π}{6}}^{\frac{π}{3}} 1 dx = {[x]}_{\frac{π}{6}}^{\frac{π}{3}} = \frac{π}{3} - \frac{π}{6} = \frac{π}{6} \Rightarrow I = \frac{π}{12}$