Problems on Techniques of Integration

The exponential function is a nice function since its derivative and antiderivative are the same. So for the integration by parts, this function has the same behavior whether we differentiate it or take its antiderivative. Therefore the focus should be on the other function involved in the integration. In this case, we must differentiate $f(x) = x$ because its derivative gives the constant 1.

Set


\begin{displaymath}\left\{\begin{array}{lll}
u &=&x\\
dv &=& e^{2 x}dx\;.
\end{array}\right.\end{displaymath}

Then

\begin{displaymath}\left\{\begin{array}{lll}
du &=&dx\\
v &=& \displaystyle \frac{1}{2}e^{2x}\;.
\end{array}\right.\end{displaymath}

Since

\begin{displaymath}\int u dv = u v - \int v du\;,\end{displaymath}

we get

\begin{displaymath}\int xe^{x} dx = x \frac{1}{2} e^{2 x} - \int \frac{1}{2} e^{2 x} dx \end{displaymath}

which implies

\begin{displaymath}\int x e^{x} dx = \frac{x}{2} e^{2 x} - \frac{1}{4}e^{2 x} + C = \frac{1}{4}(2x-1)e^{2x} + C\;.\end{displaymath}

It is a common mistake to forget the constant $C$.


If you prefer to jump to the next problem, click on Next Problem below.

[Next Problem] [Matrix Algebra]
[Trigonometry] [Calculus]
[Geometry] [Algebra]
[Differential Equations] [Complex Variables]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Mohamed A. Khamsi

Copyright © 1999-2024 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour