Integrate the function- - xe x -1- x 2 dx

Let I =∫xe^x/(1+x)^2dx =∫ e^x (x+1) 1/(1+x)^2dx =∫ e^x [ 1/(1+x) 1/(1+x)^2 ]dx Let f(x)=1/1+x⇒ f'(x)= 1/(1+x)^2 We know that ∫ e^x (f(x)+f'(x))dx =e^x f(x) ⇒ I ...

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Standard XII

Mathematics

Quotient Rule of Differentiation

Integrate the...

Question

Integrate the function.
$\int \frac{x e^{x}}{(1 + x)^{2}} d x .$

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Solution

Let $I = \int \frac{x e^{x}}{(1 + x)^{2}} d x = \int e^{x} \frac{(x + 1) - 1}{(1 + x)^{2}} d x$
$= \int e^{x} [\frac{1}{(1 + x)} - \frac{1}{(1 + x)^{2}}] d x$
Let $f (x) = \frac{1}{1 + x} \Rightarrow f^{'} (x) = - \frac{1}{(1 + x)^{2}}$
We know that $\int e^{x} (f (x) + f^{'} (x)) d x = e^{x} f (x)$
$\Rightarrow I = \int e^{x} {\frac{1}{1 + x} - \frac{1}{(1 + x)^{2}}} d x = \frac{e^{x}}{1 + x} + C$

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Integrate the function. ∫xex(1+x)2dx.

Let I=∫xex(1+x)2dx=∫ex(x+1)−1(1+x)2dx =∫ex[1(1+x)−1(1+x)2]dx Let f(x)=11+x⇒f′(x)=−1(1+x)2 We know that ∫ex(f(x)+f′(x))dx=exf(x) ⇒I=∫ex{11+x−1(1+x)2}dx=ex1+x+C

Integrate the function.
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