Question
Evaluate the following integrals:
$\int\text{x}^2\tan^{-1}\text{x dx} $
$\int\text{x}^2\tan^{-1}\text{x dx} $
✓
Answer
Let $\text{I}=\int\text{x}^2\tan^{-1}\text{x dx}$
$=\tan^{-1}\text{x}\int\text{x}^2\text{dx}-\int\Big(\frac{1}{1+\text{x}^2}\int\text{x}^2\text{dx}\Big)$
$=\tan^{-1}\text{x}\Big(\frac{\text{x}^3}{3}\Big)-\frac{1}{3}\frac{\text{x}^3}{1+\text{x}^2}\text{dx}$
$=\frac{1}3{\text{x}^3}\tan^{-1}\text{x}-\frac{1}{3}\int\Big(\text{x}-\frac{\text{x}}{1+\text{x}^2}\Big)\text{dx}$
$=\frac{1}{3}\text{x}^3\tan^{-1}\text{x}-\frac{1}{3}\times\frac{\text{x}^2}{2}+\frac{1}3{}\int\frac{\text{x}}{1+\text{x}^2}\text{dx}$
$\text{I}=\frac{1}{3}\text{x}^3\tan^{-1}\text{x}-\frac{1}{6}\text{x}^2+\frac{1}{6}\log\big|1+\text{x}^2\big|+\text{C}$
$=\tan^{-1}\text{x}\int\text{x}^2\text{dx}-\int\Big(\frac{1}{1+\text{x}^2}\int\text{x}^2\text{dx}\Big)$
$=\tan^{-1}\text{x}\Big(\frac{\text{x}^3}{3}\Big)-\frac{1}{3}\frac{\text{x}^3}{1+\text{x}^2}\text{dx}$
$=\frac{1}3{\text{x}^3}\tan^{-1}\text{x}-\frac{1}{3}\int\Big(\text{x}-\frac{\text{x}}{1+\text{x}^2}\Big)\text{dx}$
$=\frac{1}{3}\text{x}^3\tan^{-1}\text{x}-\frac{1}{3}\times\frac{\text{x}^2}{2}+\frac{1}3{}\int\frac{\text{x}}{1+\text{x}^2}\text{dx}$
$\text{I}=\frac{1}{3}\text{x}^3\tan^{-1}\text{x}-\frac{1}{6}\text{x}^2+\frac{1}{6}\log\big|1+\text{x}^2\big|+\text{C}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the vector equation of the plane passing through the points A(1, -2, 1), B (2, -1, -3) and C (0, 1, 5).
Evaluate the following integrals:$\int^\limits1_{-1}5\text{x}^4\sqrt{\text{x}^5+1}\text{ dx}$
→Evaluate the following definite integrals:$\int_{1}^\limits{2}\text{e}^{2\text{x}}\Big(\frac{1}{\text{x}}-\frac{1}{2\text{x}^2}\big)\text{dx}$
→A random variable X has the following probability distribution:
Determine:
$\text{P}(\text{X}<3),\text{P}(\text{X}\geq3),\text{P}(0<\text{X}<5).$
→| Values of X: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| P(X) | a | 3a | 5a | 7a | 9a | 11a | 13a | 15a | 17a |
$\text{P}(\text{X}<3),\text{P}(\text{X}\geq3),\text{P}(0<\text{X}<5).$
If $\int\limits^{\text{a}}_0\frac{1}{4+\text{x}^2}\text{ dx}=\frac{\pi}{8},$ find the value of a.
→In ∆ABC, if ∠A = 45º, ∠B = 60º then find the ratio of its sides.
Find the approximate values of:
→$\sin \left(29^{\circ} 30^{\circ}\right)$, given that $1^{\circ}=0.0175^{\circ}, \sqrt{ } 3=1.732$.
Find the expected value, variance, and standard deviation of the random variable whose p.m.f’s are given below:
Prove that $f(x) = ax + b$, where a, b are constants and $a > 0$ is an increasing function on R.
→Find the condition that the equation $a y^2+b x y+e x+d y=0$ may represent a pair of lines.
→