What is the integral of ((t^2+1)e^t)/((t+1)^2) ?

The integral of ((t^2+1)e^t)/((t+1)^2) is -(e^t(t^2+1))/(t+1)+e^tt+C

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integral of ((t^2+1)e^t)/((t+1)^2)

\int \frac{( t ^{2} + 1 ) e ^{t}}{( t + 1 ) ^{2}} d t

- \frac{e ^{t} ( t ^{2} + 1 )}{t + 1} + e^{t} t + C

Solution steps

\int \frac{( t ^{2} + 1 ) e ^{t}}{( t + 1 ) ^{2}} d t

Apply Integration By Parts

: - \frac{e ^{t} ( t ^{2} + 1 )}{t + 1} - \int - e^{t} (t + 1) d t

= - \frac{e ^{t} ( t ^{2} + 1 )}{t + 1} - \int - e^{t} (t + 1) d t

\int - e^{t} (t + 1) d t = - e^{t} t

= - \frac{e ^{t} ( t ^{2} + 1 )}{t + 1} - (- e^{t} t)

Simplify

= - \frac{e ^{t} ( t ^{2} + 1 )}{t + 1} + e^{t} t

Add a constant to the solution

= - \frac{e ^{t} ( t ^{2} + 1 )}{t + 1} + e^{t} t + C

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What is the integral of ((t^2+1)e^t)/((t+1)^2) ?
The integral of ((t^2+1)e^t)/((t+1)^2) is -(e^t(t^2+1))/(t+1)+e^tt+C