Answer
$ \displaystyle \frac{e^{(x+1)^{2}}}{2}+C$
Work Step by Step
see Substitution RuIe, p.962:
1. Write $u$ a{\it s} a function of x.
2. Take the derivative $du/dx$ and solve for the quantity $dx$ in terms of $du$.
3. Use the expression you obtain in step 2 to substitute for $dx$ in the given integral and substitute $u$ for its defining expression.
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(1) Given $\quad u=(x+1)^{2}$
(2)$ \displaystyle \quad du=2(x+1)dx\ \ \Rightarrow\ \ (x+1)dx=\frac{du}{2}$
$(dx=\displaystyle \frac{du}{2(x+1)})$
$(3)$
$\displaystyle \int(x+1)e^{(x+1)^{2}}dx-\int e^{(x+1)^{2}}[(x+1)dx]$
$=\displaystyle \int e^{u}(\frac{du}{2})$= ... constant multiple
$=\displaystyle \frac{1}{2}\int e^{u}du$
$=\displaystyle \frac{e^{u}}{2}+C$= ... bring back x
$= \displaystyle \frac{e^{(x+1)^{2}}}{2}+C$
