Exx2-1-x-12dx is equal to

The correct option is B (x 1/x+1 )e^x+c Consider, I=∫e^x(x^2+1)/(x+1)^2dx =∫e^x(x^2 1+2)/(x+1)^2dx =∫ e^x [ x 1/x+1+2/(x+1)^2 ]dx =∫ e ...

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Byju's Answer

Standard VI

Mathematics

Idea of a Set

exx2+1/x+12dx...

Question

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

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

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

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e^{x} (x + 1) (x - 1) + c

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none of these

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Solution

The correct option is B $(\frac{x - 1}{x + 1}) e^{x} + c$
Consider, $I = \int \frac{e^{x} (x^{2} + 1)}{(x + 1)^{2}} d x$
$= \int \frac{e^{x} (x^{2} - 1 + 2)}{(x + 1)^{2}} d x$
$= \int e^{x} [\frac{x - 1}{x + 1} + \frac{2}{(x + 1)^{2}}] d x$
$= \int e^{x} [f (x) + f^{'} (x)] d x,$
where $f (x) = \frac{x - 1}{x + 1}$
and $f^{'} (x) = \frac{2}{(x + 1)^{2}}$
So, $I = e^{x} (\frac{x - 1}{x + 1}) + C$

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ex(x2+1)(x+1)2dx is equal to

The correct option is B (x−1x+1)ex+cConsider, I=∫ex(x2+1)(x+1)2dx=∫ex(x2−1+2)(x+1)2dx=∫ex[x−1x+1+2(x+1)2]dx=∫ex[f(x)+f′(x)]dx,where f(x)=x−1x+1and f′(x)=2(x+1)2So, I=ex(x−1x+1)+C

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