Mathematics

Simplify: $$\int {\dfrac{{{e^x}\left( {x - 1} \right)}}{{{{\left( {x + 1} \right)}^3}}}dx} $$


SOLUTION
$$I=\int e^x\frac{(x-1)}{(x+1)^3}dx $$
$$I=\int e^x\frac{(x+1-2)}{(x+1)^3}dx $$
$$I=\int e^x\frac{1+x}{(x+1)^3}dx $$ $$-$$ $$\int e^x\frac{2}{(1+x)^3}dx $$ 
$$I=\int e^x\frac{1}{(1+x)^2}dx $$ $$-$$ $$\int e^x\frac{2}{(x+1)^3}dx $$ 
By integration by parts : $$\int [f(x) g(x)]dx=f(x)\cdot \int g(x)dx-\int[f'(x) \int g(x) dx]dx$$
By integration by  parts of first integral
$$I=$$ $$\frac { 1 }{ (1+ x )^{ 2 } } \int { e } ^{ x }-\int { \{ (\frac { d(\frac { 1 }{ (1+ x )^{ 2 } } ) }{ dx }  } )\int { { e }^{ x } } dx\} dx$$   $$-$$ $$\int e^x\frac{2x}{(1+x^2)^2}dx $$
$$I= e^x\frac{1}{(1+x)^2} $$ $$+$$ $$\int e^x\frac{2}{(1+x)^3}dx $$ $$-$$ $$\int e^x\frac{2}{(1+x)^3}dx $$
$$I= e^x\frac{1}{(1+x)^2} $$ $$+$$ $$C$$
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Subjective Medium Published on 17th 09, 2020
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