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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

\int - e^{x} d x = - e^{x}

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

Simplify

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

Add a constant to the solution

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

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