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.
$\displaystyle \int{u}dv$ $\,=\,$ $uv$ $-$ $\displaystyle \int{v}du$
$\implies$ $\displaystyle \int{xe^x \,}dx$ $\,=\,$ $x \times e^x$ $-$ $\displaystyle \int{e^x \,}dx$
$\implies$ $\displaystyle \int{xe^x \,}dx$ $\,=\,$ $xe^x$ $-$ $\displaystyle \int{e^x \,}dx$
$\implies$ $\displaystyle \int{xe^x \,}dx$ $\,=\,$ $xe^x-e^x+c$
$\,\,\, \therefore \,\,\,\,\,\,$ $\displaystyle \int{xe^x \,}dx$ $\,=\,$ $e^x(x-1)+c$
A best free mathematics education website for students, teachers and researchers.
Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.
Learn how to solve the maths problems in different methods with understandable steps.
Copyright © 2012 - 2022 Math Doubts, All Rights Reserved