Evaluate:

Question:

Evaluate: $\int\left(e^{x}+1\right)^{2} e^{x} d x$

Solution:

Let $I=\int\left(e^{x}+1\right)^{2} e^{x} d x$

Let $\mathrm{e}^{\mathrm{x}}+1=\mathrm{t}=\mathrm{e}^{\mathrm{x}} \mathrm{dx}=\mathrm{dt}$

$I=\int\left(e^{x}+1\right)^{2} e^{x} d x$

$=\int t^{2} d t$

$=\frac{t^{3}}{3}$

Now, substitute the value of $t$

Hence, $I=\frac{\left(e^{x}+1\right)^{3}}{3}+C$

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