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Integrals

Multiple Choice Questions

441.

$\int \frac{e^{x}}{(2 + e^{x}) (e^{x} + 1)} dx$ is equal to

$\log (\frac{e^{x} + 1}{e^{x} + 2}) + c$
$\log (\frac{e^{x} + 2}{e^{x} + 1}) + c$
$(\frac{e^{x} + 1}{e^{x} + 2}) + c$
$(\frac{e^{x} + 2}{e^{x} + 1}) + c$

442.

$\int 32 x^{3} {(\log (x))}^{2} dx$ is equal to

8x⁴(log(x))² + c
$x^{4} \{8 {(\log (x))}^{2} - 4 (\log (x)) + 1\} + c$
$\{8 {(\log (x))}^{2} - 4 (\log (x))\} + c$
$x^{3} \{8 {(\log (x))}^{2} - 2 (\log (x))\} + c$

443.

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

$\frac{e^{x} \cos (x)}{1 + \sin (x)} + c$
$c - \frac{e^{x} \sin (x)}{1 + \sin (x)}$
$c - \frac{e^{x}}{1 + \sin (x)}$
$c - \frac{e^{x} \cos (x)}{1 + \sin (x)}$

444.

If $\int f (x) dx = g (x) + c, then \int f^{- 1} (x) dx$ is equal to :

xf^-1(x) + c
f(g^-1(x)) + c
xf^-1(x) - g(f^-1(x)) + c
g^-1(x) + c

445.

The value of $\int_{1}^{2} \frac{dx}{x (1 + x^{4})}$ is :

$\frac{1}{4} \log (\frac{17}{32})$
$\frac{1}{4} \log (\frac{32}{17})$
$\log (\frac{17}{2})$
$\frac{1}{4} \log (\frac{17}{2})$

446.

The value of the integral $\int_{a}^{b} \frac{\sqrt{x} dx}{\sqrt{x} + \sqrt{a + b - x}}$ is :

$π$
$\frac{1}{2} (b - a)$
$π / 2$
b - a

447.

$\int_{0}^{\frac{π}{2}} \frac{\sqrt{\cot (x)}}{\sqrt{\cot (x)} + \sqrt{\tan (x)}} dx$ is equal to :

1
- 1
$\frac{π}{2}$
$\frac{π}{4}$

448.
$\int_{0}^{\frac{π}{2}} {xsin}^{2} (x) \cos^{2} (x) dx$ is equal to :

$\frac{π^{2}}{32}$

$\frac{π^{2}}{16}$

$\frac{π}{32}$

None of these

$\frac{π^{2}}{32}$

$Let I = \int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} {xsin}^{2} (x) \cos^{2} (x) dx . . . (i) I = \int_{0}^{\frac{π}{2}} (\frac{π}{2} - x) \sin^{2} (x) \cos^{2} (x) dx . . . (ii) Adding Eqs . (i) and (ii) we get 2 I = \frac{π}{2} \int_{0}^{\frac{π}{2}} \sin^{2} (x) \cos^{2} (x) dx = \frac{π}{8} \int_{0}^{\frac{π}{2}} \sin^{2} (2 x) = \frac{π}{8} \int_{0}^{\frac{π}{2}} (1 - \cos (4 x)) dx \Rightarrow 2 I = \frac{π}{8} {[x - \frac{\sin (4 x)}{4}]}_{0}^{\frac{π}{2}} = \frac{π}{8} [\frac{π}{2}] = \frac{π^{2}}{32}$