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Integrals

Multiple Choice Questions

111.

If for every integer n, $\int_{n}^{n + 1} f (x) d x = n^{2}$ , then the value of $\int_{- 2}^{4} f (x) d x$ is

16
14
19
None of these

112.

The value of integral $\int_{0}^{π} xf (\sin (x)) d x$ is

0
$π \int_{0}^{\frac{π}{2}} f (\sin (x)) d x$
$\frac{π}{4} \int_{0}^{π} f (\sin (x)) d x$
None of these

113.

$\lim_{x \to \infty} \frac{\int_{0}^{2 x} {xe}^{x} dx}{e^{4 x^{2}}}$ equals

0
$\infty$
2
$\frac{1}{2}$

114.

The value of $\int_{- 2}^{3} |1 - x^{2}| d x$ , is

$\frac{1}{3}$
14/3
7/3
28/3

115.

$\int \frac{1}{\tan (x) + \cot (x) + \sec (x) + \csc (x)} dx$ is equal to

$\frac{1}{2} (\sin (x) + \cos (x) + x)$ + C
$\frac{1}{2} (\sin (x) - \cos (x) - x)$ + C
$\frac{1}{2} (\cos (x) - x + \sin (x))$ + C
None of the above

116.

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 $π$

117.

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

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

118.

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

119.
$\int_{\frac{5}{2}}^{5} \frac{\sqrt{{(25 - x^{2})}^{3}}}{x^{4}} dx$ is equal to

$\frac{π}{3}$

$\frac{2 π}{3}$

$\frac{π}{6}$

$\frac{5 π}{6}$

$\frac{π}{3}$

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

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