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 (\(\frac {x^2}{2}\))-\(∫\frac {1}{1+x^2}.\frac {x^2}{2} dx\)
\(I\)= \(\frac {x^2tan^{-1}x}{2}\) - \(\frac 12\)\(∫\frac {x^2}{1+x^2} dx\)
\(I\)= \(\frac {x^2tan^{-1}x}{2}\) - \(\frac 12\)\(∫(\frac {x^2+1}{1+x^2}-\frac {1}{1+x^2})dx\)
\(I\)= \(\frac {x^2tan^{-1}x}{2}\) - \(\frac 12\)\(∫(1-\frac {1}{1+x^2})dx\)
\(I\)= \(\frac {x^2tan^{-1}x}{2}\) - \(\frac 12\)\((x-tan^{-1}x)+C\)
\(I\)= \(\frac {x^2}{2}tan^{-1}x - \frac x2+\frac 12tan^{-1}x+C\)
Top Questions on integral
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Questions Asked in CBSE CLASS XII exam
- Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 1 & 3\\ 2 & 7\end{bmatrix}\)
- For what values of x,\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 & 0\\ 2 & 0 & 1 \\1&0&2 \end{bmatrix}\)\(\begin{bmatrix} 0 \\2\\x\end{bmatrix}\)=O?
- Find the inverse of each of the matrices,if it exists \(\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}\)
What is the Planning Process?
- CBSE CLASS XII - 2023
- Planning process steps
- Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)
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,
