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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

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

$Let I = \int f^{- 1} (x) dx . . . (i) and \int f (x) dx = g (x) + c . . . (ii) ∴ From (i) let f^{- 1} (x) = u \Rightarrow x = f (u) \Rightarrow dx = f' (u) du ∴ I = \int uf' (u) du Integration by parts, we get I = uf (u) - \int f (u) du I = uf (u) - g (u) + c [∵ usin Eq . (ii)] ∴ On putting u = f^{- 1} (x), f (u) = x We get, I = x f^{- 1} (x) - g (f^{- 1} (x)) + c$