Integrate the function: \(xtan^{-1}x\)
Solution and Explanation
Let \(I\) =\(∫\)\(xtan^{-1}x\ dx\)
Taking as first function and x as second function and integrating by parts, we obtain
\(I\) = tan-1x∫x dx-∫{(\(\frac {d}{dx}\)tan-1x)∫x dx}dx
\(I\)= tan-1x (\(\frac {x^2}{2}\))-\(∫\frac {1}{1+x^2}.\frac {x^2}{2} dx\)
\(I\)= \(\frac {x^2tan^{-1}x}{2}\) - \(\frac 12\)\(∫\frac {x^2}{1+x^2} dx\)
\(I\)= \(\frac {x^2tan^{-1}x}{2}\) - \(\frac 12\)\(∫(\frac {x^2+1}{1+x^2}-\frac {1}{1+x^2})dx\)
\(I\)= \(\frac {x^2tan^{-1}x}{2}\) - \(\frac 12\)\(∫(1-\frac {1}{1+x^2})dx\)
\(I\)= \(\frac {x^2tan^{-1}x}{2}\) - \(\frac 12\)\((x-tan^{-1}x)+C\)
\(I\)= \(\frac {x^2}{2}tan^{-1}x - \frac x2+\frac 12tan^{-1}x+C\)
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Concepts Used:
Integration by Partial Fractions
The number of formulas used to decompose the given improper rational functions is given below. By using the given expressions, we can quickly write the integrand as a sum of proper rational functions.
For examples,
