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Integrals

Multiple Choice Questions

401.

If $\int_{0}^{\infty} \frac{\log (1 + x^{2})}{1 + x^{2}} d x = k \int_{0}^{1} \frac{\log (1 + x)}{1 + x^{2}} dx$ , then k is equal to

4
8
$π$
2 $π$

402.

$\int_{0}^{x} |\sin (t)| dt, where x \in [2 nπ, (2 n + 1) π], n \in N$ , is equal to

4n - 1 - cos(x)
4n - sin(x)
4n - cos(x)
4n + 1 - cos(x)

403.

Let I₁ = $\int_{0}^{1} \frac{e^{x} dx}{1 + x}$ and I₂ = $\int_{0}^{1} \frac{x^{2} dx}{e^{x^{3}} (2 - x^{3})}$ Then, $\frac{I_{1}}{I_{2}}$ is equal to

$\frac{1}{3 e}$
3e
$\frac{e}{3}$
$\frac{3}{e}$

404.
$\int_{\frac{5}{2}}^{5} \frac{\sqrt{{(25 - x^{2})}^{3}}}{x^{4}} dx$ is equal to

$\frac{π}{3}$

$\frac{2 π}{3}$

$\frac{π}{6}$

$\frac{5 π}{6}$

$\frac{π}{3}$

$I = \int_{\frac{5}{2}}^{5} \frac{\sqrt{{(25 - x^{2})}^{3}}}{x^{4}} dx Let x = 5 \sin (θ) \Rightarrow dx = 5 \cos (θ) dθ ∴ I = \int_{\frac{π}{6}}^{\frac{π}{2}} \frac{\sqrt{{(25 - 25 \sin^{2} (θ))}^{3}}}{x^{4} \sin^{4} (θ)} . 5 \cos (θ) dθ$

$= \int_{\frac{π}{6}}^{\frac{π}{2}} \frac{5^{3} \cos^{3} (θ) . 5 \cos (θ)}{5^{4} \sin^{4} (θ)} = \int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (θ) (\csc^{2} (θ) - 1) dθ = \int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (θ) \csc^{2} (θ) dθ - \int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (θ) = \int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (θ) \csc^{2} (θ) dθ - \int_{\frac{π}{6}}^{\frac{π}{2}} (\csc^{2} (θ) - 1) dθ = {[- \frac{\cot^{3} (θ)}{3} + \cot (θ) + θ]}_{\frac{π}{6}}^{\frac{π}{2}} = (- 0 + 0 + \frac{π}{2}) - (- \frac{3 \sqrt{3}}{3} + \sqrt{3} + \frac{π}{6}) = \frac{π}{3}$

405.

$\int \frac{dx}{\cos (x) + \sqrt{3} \sin (x)}$ equals

$\frac{1}{2} \log (\tan (\frac{x}{2} + \frac{π}{12})) + C$
$\frac{1}{3} \log (\tan (\frac{x}{2} - \frac{π}{12})) + C$
$\log (\tan (\frac{x}{2} + \frac{π}{6})) + C$
$\frac{1}{2} \log (\tan (\frac{x}{2} - \frac{π}{6})) + C$

406.

$\int_{0}^{\frac{π}{2}} \sin (2 x) . \log (\tan (x)) dx$

407.

If $∆ x = |\begin{matrix} 1 & \cos (x) & 1 - \cos (x) \\ 1 + \sin (x) & \cos (x) & 1 + \sin (x) + \cos (x) \\ \sin (x) & \sin (x) & 1 \end{matrix}|$ , then $\int_{0}^{\frac{π}{4}} ∆ (x) dx$ is equal to