Question

Find : $\int \frac{e^{x}(1-x^{2})}{(1+x^{2})^{2}}dx$​

Answer 1

Answer :

$\int\frac{(1 - x^{2})e^{x}}{(1 + x^{2})^{2}}\: dx \\ \\ \int\frac{(1 - x^{2} - 2x)e^{x}}{(1 + x^{2})^{2}}\: dx \\ \\ \int \frac{e^{x}}{(1 + x ^{2})} dx + \int \frac{ - 2x \: e^{x}}{(1 + x ^{2})^{2} } dx$

Now, tackling the integer on the left first,

$\int \frac{e^{x}}{(1 + x ^{2})} dx$

Integration of parts :

$u = (1 + x {}^{2} )^{ - 1} ; \\ du = - 2x(1 + x^{2})^{ - 2} dx \\ \\ dv = e^{x} dx \: ; \: v = e^{x}$

And this remaining integral cancels -

$\boxed{ \bf \frac{e^{x} }{(1 + x {}^{2})} + C }$

Answer 2

$\LARGE{\red{\boxed{\purple{\underline{\orange{\mathtt{Question:↓}}}}}}}$

Find:

$\int \frac{e^{x}(1-x^{2})}{(1+x^{2})^{2}}dx$

$\LARGE{\red{\boxed{\purple{\underline{\orange{\mathtt{Answer:↓}}}}}}}$

$\begin{gathered} \boxed{ \bf \frac{e^{x} }{(1 + x {}^{2})} + C }\\\end{gathered}$

$\begin{gathered} \int\frac{(1 - x^{2})e^{x}}{(1 + x^{2})^{2}}\: dx \\ \\ \int\frac{(1 - x^{2} - 2x)e^{x}}{(1 + x^{2})^{2}}\: dx \\ \\ \int \frac{e^{x}}{(1 + x ^{2})} dx + \int \frac{ - 2x \: e^{x}}{(1 + x ^{2})^{2} } dx \end{gathered}$

Now, tackling the integer on the left first,

$\begin{gathered}\int \frac{e^{x}}{(1 + x ^{2})} dx \\ \end{gathered}$

Integration of parts :

$\begin{gathered}u = (1 + x {}^{2} )^{ - 1} ; \\ du = - 2x(1 + x^{2})^{ - 2} dx \\ \\ dv = e^{x} dx \: ; \: v = e^{x}\end{gathered}$

And this remaining integral cancels -

$\begin{gathered} \boxed{ \bf \frac{e^{x} }{(1 + x {}^{2})} + C }\\\end{gathered}$