Evaluate: _0^ 1 / 2 ^ -1 x (1-x^2 )^ 3 / 2 d x - Vidyadip

Question

Evaluate: $\int_0^{1 / \sqrt{2}} \frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}} d x$

✓

Answer

Let $\sin ^{-1} x=\theta$ or, $x=\sin \theta$. Then, $d x=d(\sin \theta)=\cos \theta d \theta$
Now, $ x=0$
$\Rightarrow \sin \theta=0$
$\Rightarrow \theta=0$ and $x=\frac{1}{\sqrt{2}}$
$\Rightarrow \sin \theta=\frac{1}{\sqrt{2}} $$\Rightarrow \theta=\frac{\pi}{4}$
$\therefore I=\int_0^{1 / \sqrt{2}} \frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}} d x$
$\Rightarrow I=\int_0^{\pi / 4} \frac{\theta}{\cos ^3 \theta} \cos \theta d \theta=\int_0^{\pi / 4} \theta \sec ^2 \theta d \theta$
Now using integratio by parts.
$\Rightarrow I=\left[\theta \tan \theta\right]_0^{\pi / 4}+[\log \cos \theta]_0^{\pi / 4}=\frac{\pi}{4}+\left\{\log \left(\frac{1}{\sqrt{2}}\right)-\log 1\right\}=\frac{\pi}{4}-\frac{1}{2} \log 2$

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 Model Paper 5→Model Paper 5 — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

By using the properties of definite integrals, evaluate the integral $\int\limits_0^2 {x\sqrt {2 - x} dx} $

Find the intervals in which the following functions are increasing or decreasing. $f(x) = 6+ 12x + 3x^2 - 2x^3$

Dot product of a vector with vectore $\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}},2\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}$ and $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ are respectively 4, 0 and 2. Find the vector.

Find the points o local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:
$\text{f}(\text{x})=\text{x}\sqrt{1-\text{x}}, \text{x}\geq0$

Let $A = [-1, 1].$ Then, discuss whether the following functions from $A$ to itself are one$-$one, onto or bijective:$h(x) = x^2$

Give an example of a function which is continuos but not differentiable at at a point.

Write the following function in the simplest form:
$\tan^{-1}\bigg(\frac{3a^{2}x-x^{3}}{a^{3}-3ax^{2}}\bigg), a>0; \frac{-a}{\sqrt{3}}\leq x\leq\frac{a}{\sqrt{3}}$

Evaluate the following integrals:
$\int\text{x}^2\sqrt{\text{a}^6-\text{x}^6}\text{dx}$

A laboratory blood test is $99\%$ effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for $0.5\%$ of the healthy person tested $($i.e. if a healthy person is tested, then, with probability $0.005,$ the test will imply he has the disease$).$ If $0.1$ percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

Write the following function in the simplest form:
$\tan^{-1}\frac{x}{\sqrt{a^{2}-x^{2}}}, \left|x\right|