What is the integral of xe^{ln(x+1)} ?

The integral of xe^{ln(x+1)} is (x^3)/3+(x^2)/2+C

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integral of xe^{ln(x+1)}

\int x e^{l n (x + 1)} d x

\frac{x ^{3}}{3} + \frac{x ^{2}}{2} + C

Solution steps

\int x e^{l n (x + 1)} d x

e^{l n (x + 1)} = x + 1

= \int x (x + 1) d x

Expand

x (x + 1) : x^{2} + x

= \int x^{2} + x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} d x + \int x d x

\int x^{2} d x = \frac{x ^{3}}{3}

\int x d x = \frac{x ^{2}}{2}

= \frac{x ^{3}}{3} + \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + C

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What is the integral of xe^{ln(x+1)} ?
The integral of xe^{ln(x+1)} is (x^3)/3+(x^2)/2+C