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

Question

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

✓

Answer

$\int\frac{\text{dx}}{\sqrt{1-\text{x}^2}(\sin^{-1}\text{x})^2}$
$\text{Let,}\sin^{-1}\text{x}=\text{t}$
$\Rightarrow\frac{1}{\sqrt{1-\text{x}^2}}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\frac{1}{\sqrt{1-\text{x}^2}}\text{dx}=\text{dt}$
$\text{Now,}\int\frac{\text{dx}}{\sqrt{1-\text{x}^2}(\sin^{-1}\text{x})^2}$
$=\int\frac{\text{dt}}{\text{t}^2}$
$=\int\text{t}^{-2}\text{dt}$
$=\frac{\text{t}^{-2+1}}{-2+1}+\text{C}$
$=\frac{-1}{\text{t}}+\text{C}$
$=-\frac{1}{\sin^{-1}\text{x}}+\text{C}$

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 "3 Marks Question" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A can hit a target 3 times in 6 shots, B : 2 times in 6 shots and C : 4 times in 4 shots. They fix a volley. What is the probability that at least 2 shots hit?

In a bank, the principal increases continuously at the rate of $r\%$ per year. Find the value of r if Rs.$100$ double itself in $10$ years $(log_e2 = 0.6931).$

Differentiate the function $(\log x)^{\log x}, x > 1,$ w.r.t to x.

Find the points on the line $\frac{\text{x}+2}{3}=\frac{\text{y}+1}{2}=\frac{\text{z}-3}{2}$ at a distance of 5 units from the point P(1, 3, 3).

Integrate the function $\sqrt{x^{2}+3 x}$

Differentiate the function $(x+3)^{2} \cdot(x+4)^{3} \cdot(x+5)^{4}$ w.r.t. x.

Show that the Signum Function f: R → R, given by $\text{f(x)}=\begin{cases}1,&\text{if }\text{x }>0\\0,&\text{if }\text{x }=0\\-1,&\text{if }\text{x }<0\end{cases}$ is neither one-one nor onto.

Write the following in the simplest form:
$\tan^{-1}\sqrt{\frac{\text{a}-\text{x}}{\text{a}+\text{x}}},-\text{a}<\text{x}<\text{a}$

Find the rate of change of the volume of a sphere with respect to its diameter.

Evaluate the following definite integrals:
$\int_{\frac{\pi}{4}}^\limits{\frac{\pi}{2}}\cot\text{x}\text{ dx}$