Question
$\int e ^x \frac{\left(1+x^2\right)}{(1+x)^2} d x$
✓
Answer
$\text { Let } I =\int e ^x \frac{\left(1+x^2\right)}{(1+x)^2} d x$
$=\int e ^x\left[\frac{x^2-1+2}{(1+x)^2}\right] d x$
$=\int e ^x\left[\frac{x^2-1}{(x+1)^2}+\frac{2}{(x+1)^2}\right] d x$
$=\int e ^x\left[\frac{x-}{x+1}+\frac{2}{(x+1)^2}\right] d x$
Put $f ( x )=\frac{x-1}{x+1}$
$\therefore f ^{\prime}( x )=\frac{(x+1)(1-0)-(x-1)(1+0)}{(x+1)^2}$
$=\frac{2}{(x+1)^2}$
$\therefore I =\int e ^x\left[ f (x)+ f ^{\prime}(x)\right] d x$
$= e ^{ x } \cdot f ( x )+ c$
$= e ^x\left(\frac{x-1}{x+1}\right)+ c$
$=\int e ^x\left[\frac{x^2-1+2}{(1+x)^2}\right] d x$
$=\int e ^x\left[\frac{x^2-1}{(x+1)^2}+\frac{2}{(x+1)^2}\right] d x$
$=\int e ^x\left[\frac{x-}{x+1}+\frac{2}{(x+1)^2}\right] d x$
Put $f ( x )=\frac{x-1}{x+1}$
$\therefore f ^{\prime}( x )=\frac{(x+1)(1-0)-(x-1)(1+0)}{(x+1)^2}$
$=\frac{2}{(x+1)^2}$
$\therefore I =\int e ^x\left[ f (x)+ f ^{\prime}(x)\right] d x$
$= e ^{ x } \cdot f ( x )+ c$
$= e ^x\left(\frac{x-1}{x+1}\right)+ c$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If $y =\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\ldots \ldots \infty}}}$, show that $\frac{ d y}{ d x}=\frac{\sin x}{1-2 y}$
→Express the following circuits in the symbolic form of logic and writ the input-output table.
Solve the following differential equations : $x \sin \frac{d y}{d x}+(x \cos x+\sin y)=\sin x$
→Obtain the differential equation by eliminating the arbitrary constants from the following equations:
→$y=c_1 e^{2 x}+c_2 e^{5 x}$
If D is the mid-point of side BC of a triangle ABC such that $\overrightarrow{\text{AB}}+\overrightarrow{\text{AC}}=\lambda\overrightarrow{\text{AD}}$, write the value of $\lambda$.
→Find the probability of guessing correctly at least nine out of ten answers in a "true" or "false" objective test
Find the Cartesian equations of the line passing through A(-1, 2, 1) and having direction ratios 2, 3, 1.
Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
$\int \frac{x^2+x-1}{x^2+x-6} d x$
→Write the value of $\cos^{-1}\Big(\tan\frac{3\pi}{4}\Big).$
→