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" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Given $\text{A}=\begin{bmatrix}2 & -3 \\-4 & 7 \end{bmatrix},$ compute A^-1 and show that 2A^-1 = 9I - A.

Evalute the following integrals:
$\int\frac{\sec\text{x cosec x}}{\log(\tan\text{x})}\text{dx}$

Evaluate the following integrals:
$\int_{0}^\limits{1}\text{xe}^{\text{x}^2}\text{ dx}$

Find the local maxima and local minima of the function
f(x) = x³ - 6x² + 9x + 15
Find also the local maximum and the local minimum value.

Evaluate the following integrals:
$\int^\limits4_1\text{f(x)}\text{dx},$ Where $\text{f(x)}=\begin{cases}4\text{x}+3,&\text{if }\ 1\leq\text{x}\leq2\\3\text{x}+5,&\text{if }\ 2\leq\text{x}\leq4\end{cases}$

Find the second order derivatives of the following functions:

$\sin(\log\text{x})$

Integrate the function in Exercise:
$\frac{\text{x}\cos^{-1}\text{x}}{\sqrt{1-\text{x}^2}}$

Evaluate the following integrals:

$\int\frac{\log\text{x}}{\text{x}^{\text{n}}}\text{dx}$

Evaluate the following intregals:
$\int\frac{1}{\cos\text{x}(\sin\text{x}+2\cos\text{x})}\ \text{dx}$

Find the equation of a plane which meets the axes at A, B and C, given that the centroid of the triangle ABC is the point $(\alpha,\beta,\gamma)$