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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 :

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

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

$\log (\frac{17}{2})$

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

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

$Let I = \int_{1}^{2} \frac{dx}{x (1 + x^{4})} I = \int_{1}^{2} \frac{x^{3}}{x^{4} (1 + x^{4})} dx On putting x^{4} = t \Rightarrow 4 x^{3} dx = dt ∴ I = \frac{1}{4} \int_{1}^{16} \frac{dt}{t (1 + t)} = \frac{1}{4} \int_{1}^{16} (\frac{1}{t} - \frac{1}{1 + t}) dt = \frac{1}{4} \log {(\frac{t}{1 + t})}_{1}^{16} = \frac{1}{4} (\log (\frac{16}{17}) - \log (\frac{1}{2})) = \frac{1}{4} \log (\frac{32}{17})$