The variable $x$ and natural exponential function $e^x$ formed an algebraic function by their product, and we have to evaluate the indefinite integration of the function $xe^x$ with respect to $x$ in calculus.
$\displaystyle \int{xe^x \,} dx$
In this integral problem, we can notice that
The indefinite integration of the given algebraic function can be evaluated only by the integration by parts method.
Take, $\displaystyle \int{xe^x \,} dx$ $\,=\,$ $\displaystyle \int{u}dv$
In this indefinite integration problem, we use the power reduction technique for solving this problem. So, we must take $u = x$ and $dv = e^x dx$
Now, we have to evaluate the differential element $du$ by differentiation and the variable $v$ by the integration.
$u = x$
$\implies$ $\dfrac{du}{dx} = \dfrac{dx}{dx}$
$\,\,\, \therefore \,\,\,\,\,\,$ $du = dx$
$dv = e^x dx$
Now, solve the differential equation by using the integration rule of natural exponential function.
$\implies$ $\displaystyle \int{\,}dv = \int{e^x \,}dx$
$\implies$ $\displaystyle \int{\,}dv = \int{e^x \,}dx$
$\implies$ $v+c = e^x+c$
$\,\,\, \therefore \,\,\,\,\,\,$ $v = e^x$
Now, substitute the values of the variables and differentials in the formula of the integration of parts for evaluating the indefinite integration of the given algebraic function mathematically.
$\displaystyle \int{u}dv$ $\,=\,$ $uv$ $-$ $\displaystyle \int{v}du$
$\implies$ $\displaystyle \int{xe^x \,}dx$ $\,=\,$ $x \times e^x$ $-$ $\displaystyle \int{e^x \,}dx$
$\implies$ $\displaystyle \int{xe^x \,}dx$ $\,=\,$ $xe^x$ $-$ $\displaystyle \int{e^x \,}dx$
$\implies$ $\displaystyle \int{xe^x \,}dx$ $\,=\,$ $xe^x-e^x+c$
$\,\,\, \therefore \,\,\,\,\,\,$ $\displaystyle \int{xe^x \,}dx$ $\,=\,$ $e^x(x-1)+c$
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