Evaluate the following integrals: ^2 ^ -1 x dx - Vidyadip

Question

Evaluate the following integrals:$\int\text{x}^2\sin^{-1}\text{x dx}$

✓

Answer

Let $\text{I}=\int\text{x}^2\sin^{-1}\text{x dx}$
$\text{I}=\sin^{-1}\text{x}\int\text{x}^2\text{dx}-\int\Big(\frac{1}{\sqrt{1-\text{x}^2}}\int\text{x}^2\text{dx}\Big)\text{dx}$
$=\frac{\text{x}^3}{3}\sin^{-1}\text{x}-\int\frac{\text{x}^3}{3\sqrt{1-\text{x}^2}}\text{dx}$
$\text{I}=\frac{\text{x}^3}{3}\sin^{-1}\text{x}-\frac{1}{3}\text{I}_1+\text{C}_1\dots(1)$
$\text{I}_1=\int\frac{\text{x}^3}{\sqrt{1-\text{x}^2}}\text{dx}$
Let $1-\text{x}^2=\text{t}^2$
$-2\text{x dx}=2\text{t dt}$
$-\text{x dx}=\text{t dt}$
$\text{I}_1=-\int\frac{(1-\text{t}^2)\text{tdt}}{\text{t}}$
$=\int(\text{t}^2-1)\text{dt}$
$=\frac{\text{t}^3}{3}-\text{t}+\text{C}_2$
$=\frac{(1-\text{x}^2)^{\frac{3}{2}}}{3}-(1-\text{x}^2)^{\frac{1}{2}}+\text{C}_2$
Now,
$\text{I}=\frac{\text{x}^3}{3}\sin^{-1}\text{x}-\frac{1}{9}(1-\text{x}^2)^{\frac{3}{2}}+\frac{1}{3}(1-\text{x}^2)^{\frac{1}{2}}+\text{C}$

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 "5 Marks Questions" from Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following:
$\int\limits^{\frac{\pi}{2}}_0\frac{\tan\text{x dx}}{1+\text{m}^2\tan^2\text{x}}\text{dx}$

Find the vector equation of the line through the origin which is perpendicular to the plane $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})=3$

Find the shortest distance between the following pairs of parallel lines whose equations are:$\vec{\text{r}}=\big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=\big(2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}\big)+\mu\big(-\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)$

Solve the following linear programming problem by graphical method. Under the following constraints :
$
\begin{aligned}
x+2 y & \geq 10 \\
x+y & \geq 6 \\
3 x+y & \geq 8 \\
x, y & \geq 0
\end{aligned}
$
$\operatorname{minimise} Z=3 x+5 y$.

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$\text{f}(\text{x})=\sqrt{25-\text{x}^2}\text{ on }[-3,4]$

Differentiate the following functions with respect to x:
$\frac{\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}}{\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}}$

Solve the following for x and y.$\begin{bmatrix}3&-4\\9&2\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\end{bmatrix}=\begin{bmatrix}10\\2\end{bmatrix}$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=(\text{x}+\text{y}+1)^{2}$

The bag $A$ contains $8$ white and $7$ black balls while the bag $B$ contains $5$ white and $4$ black balls. One ball is randomly picked up from the bag $A$ and mixed up with the balls in bag $B.$ Then a ball is randomly drawn out from it. Find the probability that ball drawn is white.

Find the foot of perpendicular from the point (2, 3, 4) to the line $\frac{4-\text{x}}{2}=\frac{\text{y}}{6}=\frac{1-\text{z}}{3}.$ Also, find the perpendicular distance from the given point to the line.