Question

integration of ((x²+1)e^x)/(x+1)²​

Answer 1

Step-by-step explanation:

the answer is mentioned above please check it

Answer 2

Given :

• $\sf \dfrac{((x^2+1)e^x)}{(x+1)^2}$

Solution :

$\displaystyle{\Rightarrow \sf \int \dfrac{(x^2+1)e^x}{(x+1)^2}\:dx}$

$\displaystyle{\Rightarrow \sf \int e^x\Bigg(\dfrac{x^2+1}{(x+1)^2}\Bigg)\:dx}$

$\boxed{\sf x+1=1 \Rightarrow x = t -1 \Rightarrow dx = dt}$

$\displaystyle{\Rightarrow \sf \int e^{t-1}\Bigg(\dfrac{(t-1)^2+1}{t^2}\Bigg)\:dt}$

$\displaystyle{\Rightarrow \sf \dfrac{1}{e}\int e^t \Bigg(\dfrac{t^2-2t+1+1}{t^2}\Bigg)dt}$

$\displaystyle{\Rightarrow \sf \dfrac{1}{e} \int e^t \Bigg(\dfrac{t^2-2t+2}{t^2}\Bigg)dt}$

$\displaystyle{\Rightarrow \sf \dfrac{1}{e} \Bigg(\int e^tdt-2\int \dfrac{e^t}{t}\:dt +2 \int \dfrac{e^t}{t^2}\:dt\Bigg)}$

$\displaystyle{\Rightarrow\sf \dfrac{1}{e}\Bigg(e^t -2 \Bigg[\dfrac{1}{t}\int e^tdt - \int \dfrac{d(t)}{dt}\int \Big(e^tdt\Big)dt\Bigg]+ 2 \int \dfrac{e^t}{t^2}\: dt\Bigg)}$

$\displaystyle{\Rightarrow\sf \dfrac{1}{e}\Bigg(e^t - \dfrac{2}{t}e^t+2\int \Big(\dfrac{-1}{t^2}\Big)e^tdt+2\int \dfrac{e^t}{t^2}\: dt \Bigg)}$

$\displaystyle{\Rightarrow \sf e^{t-1}-\dfrac{2}{t}e^{t-1}+C}$

$\displaystyle{\Rightarrow \sf e^x - \dfrac{2e^x}{(x+1)}+C}$

Required Answer :

$\boxed{\boxed{\sf \dfrac{(x^2+1)e^x}{(x+1)^2}= e^x - \dfrac{2e^x}{(x+1)}+C }}$