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
- Evaluate the integral: \[ \int \sqrt{x^2 + 3x} \, dx \]
- Evaluate the integral: \[ \int \frac{x^2 + 2x}{\sqrt{x^2 + 1}} \, dx \]
- The value of the integral \( \int_0^1 x^2 \, dx \) is:
- The value of the definite integral \( \int_0^{\pi} \sin^2 x \, dx \) is:
- Let $ f(x) $ be a positive function and $I_1 = \int_{-\frac{1}{2}}^1 2x \, f\left(2x(1-2x)\right) dx$ and $I_2 = \int_{-1}^2 f\left(x(1-x)\right) dx.$ Then the value of $\frac{I_2}{I_1}$ is equal to ____
Questions Asked in CBSE CLASS XII exam
- Using integration, find the area of the region bounded by the line \[ y = 5x + 2, \] the \( x \)-axis, and the ordinates \( x = -2 \) and \( x = 2 \).
- CBSE CLASS XII - 2025
- Evaluation of Definite Integrals by Substitution
- A 1 cm segment of a wire lying along the x-axis carries a current of 0.5 A along the \( +x \)-direction. A magnetic field \( \mathbf{B} = (0.4 \, \text{mT}) \hat{j} + (0.6 \, \text{mT}) \hat{k} \) is switched on in the region. The force acting on the segment is:
- CBSE CLASS XII - 2025
- Magnetic Effects of Current and Magnetism
The correct IUPAC name of \([ \text{Pt}(\text{NH}_3)_2\text{Cl}_2 ]^{2+} \) is:
- CBSE CLASS XII - 2025
- Coordination Compounds
- Two conductors A and B of the same material have their lengths in the ratio 1:2 and radii in the ratio 2:3. If they are connected in parallel across a battery, the ratio \( \frac{v_A}{v_B} \) of the drift velocities of electrons in them will be:
- CBSE CLASS XII - 2025
- Current electricity
- Arun, Bashir and Joseph were partners in a firm sharing profits and losses in the ratio of 5 : 3 : 2. They admitted Daksh as a new partner who acquired his share entirely from Arun. If Arun sacrificed \( \frac{1}{5} \)th from his share to Daksh, Daksh's share in the profits of the firm will be :
- CBSE CLASS XII - 2025
- Partnership Accounts
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,
