Evaluate the following integrals: ^3 ^ -1 x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{x}^3\tan^{-1}\text{x dx}$

✓

Answer

$\int\text{x}^3\tan^{-1}\text{x dx}$
$\int\text{x}^3\tan^{-1}\text{x dx}=\tan^{-1}\text{x}\int\text{x}^3\text{dx}-\Big(\int\frac{\text{d}\tan^{-1}\text{x}}{\text{dx}}\big(\int\text{x}^3\text{dx}\big)\text{dx}\Big)$
$=\tan^{-1}\text{x}\frac{\text{x}^4}{4}-\Big(\int\frac{1}{1+\text{x}^2}\Big(\frac{\text{x}^4}{4}\Big)\text{dx}\Big)$
$=\tan^{-1}\text{x}\frac{\text{x}^4}{4}-\Big(\int\frac{1}{1+\text{x}^2}\Big(\frac{\text{x}^4}{4}\Big)\text{dx}\Big)$
$\int\frac{1}{1+\text{x}^2}\Big(\frac{\text{x}^4}{4}\Big)\text{dx}=\frac{1}{4}\Big[\int\frac{1}{1+\text{x}^2}\text{dx}+(\text{x}^2-1)\text{dx}\Big]$
$\int\frac{1}{1+\text{x}^2}\Big(\frac{\text{x}^4}{4}\Big)\text{dx}=\frac{1}{4}\Big[\tan^{-1}\text{x}+\frac{\text{x}^3}{3}-\text{x}\Big]$
$\int\text{x}^3\tan^{-1}\text{x dx}=\frac{\text{x}^4}{4}\tan^{-1}\text{x}-\frac{1}{4}\Big[\tan^{-1}\text{x}+\frac{\text{x}^3}{3}-\text{x}\Big]+\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\frac{\text{x}^3-1}{\text{x}^3+\text{x}}\text{ dx}$

Find the area of the bounded by the curve $\text{y}=\frac{\pi}{2}+2\sin^{2}\text{x}$ x-axis and the area between x-axis, the curve and the ordinates $\text{x}=0, \text{x}=\pi$.

Show that the plane whose vector equation is $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})=3$ contains the line whose vector equation is $\vec{\text{r}}=\hat{\text{i}}+\hat{\text{j}}+\lambda(2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}}).$

Solve the following equation for x:
$\cot^{-1}\text{x}-\cot^{-1}(\text{x}+2)=\frac{\pi}{12},\text{x}>0$

Evaluate the following integrals:$\int\text{e}^{\text{x}}\frac{1+\text{x}}{(2+\text{x})^2}\text{dx}$

Given $\text{A}=\begin{bmatrix}2&2&-4\\-4&2&-4\\2&-1&5\end{bmatrix},\text{B}=\begin{bmatrix}1&-1&0\\2&3&4\\0&1&2\end{bmatrix}$ , find BA and use this to solve the system of equations $y + 2z = 7, x - y = 3, 2x + 3y + 4z = 17$

Integrate the following w.r.t. x:

$\frac{3 x+1}{\sqrt{-2 x^2+x+3}}$

Differentiate the following with respect to x:
$\cos^{-1}\Big(\frac{1-\text{x}}{1+\text{x}}\Big)$

If $\text{y}=\text{x}\sin(\text{a}+\text{y}),$ prove that $\frac{\text{dx}}{\text{dx}}=\frac{\sin^2(\text{a}+\text{y})}{\sin(\text{a}+\text{y})-\text{y}\cos(\text{a}+\text{y})}$

the cartesian equation of a line are $\frac{\text{x}-5}{3}=\frac{\text{y}+4}{7}=\frac{\text{z}-6}{2}.$ Find a vector equation for the line.