Problems on Techniques of Integration

Use the Integration by Parts. Set

u &=& \arctan(x)\\
dv &=& x dx\;.


du &=&\displaystyle \frac{1}{x^2 + 1} dx\\
v &=& \displaystyle \frac{x^2}{2}\;.


\begin{displaymath}\int x \arctan(x) dx = \frac{x^2}{2} \arctan(x) - \int \frac{...
...x^2}{2} \arctan(x) - \frac{1}{2}\int \frac{x^2}{x^2 + 1}dx\cdot\end{displaymath}


\begin{displaymath}\int \frac{x^2}{x^2 + 1}dx = \int \left(1 - \frac{1}{x^2 + 1}\right)dx = x - \arctan(x)\end{displaymath}

we get

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

Detailed Answer.

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 1999-2017 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