Integrate the function in Exercise: ^ -1 1-x 1+x - Vidyadip

Question

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

✓

Answer

$\text{I}=\tan^{-1}\sqrt{\frac{1-\text{x}}{1+\text{x}}}\text{dx}$
$\text{Let x}=\cos\theta\Rightarrow\text{dx}=-\sin\theta\text{d}\theta$
$\text{I}=\int\tan^{-1}\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}(-\sin\theta\text{d}\theta)$
$=-\int^{-1}\sqrt{\frac{2\sin^{2}\frac{\theta}{2}}{2\cos^{2}\frac{\theta}{2}}}\sin\theta\text{d}\theta$
$=-\int\tan^{-1}\tan\frac{\theta}{2}.\sin\theta\text{d}\theta$
$=-\frac{1}{2}\big[\theta.(-\cos\theta)-\int1.(-\cos\theta)\text{d}\theta\big]$
$=-\frac{1}{2}[-\theta\cos\theta+\sin\theta]$
$=+\frac{1}{2}\theta\cos\theta-\frac{1}{2}\sin\theta$
$=\frac{1}{2}\cos^{-1}\text{x}.\text{x}-\frac{1}{2}\sqrt{1-\text{x}^{2}}+\text{C}$
$=\frac{\text{x}}{2}\cos^{-1}\text{x}-\frac{1}{2}\sqrt{1-\text{x}^{2}}+\text{C}$
$=\frac{1}{2}\Big(\text{x}\cos^{-1}\text{x}-\sqrt{1-\text{x}^{2}}\Big)+\text{C}$

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

Similar questions

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

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:
$\text{f}(\text{x})=\text{x}^{3}-6\text{x}^{2}+9\text{x}+15$

Write the following function in the simplest form:
$\tan^{-1}\bigg(\frac{\cos x-\sin x}{\cos x+\sin x}\bigg), x<{\pi}$

Evaluate the following integrals:$\int\frac{\text{x}+5}{3\text{x}^2+13\text{x}-10}\text{ dx}$

In the following, find the value of the constant k so that the given function is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\frac{\text{x}^2-25}{\text{x}-5},&\text{x}\neq5\\\text{k},&\text{x}=5\end{cases}\text{at x} =5$

A laboratory blood test is $99\%$ effective in detecting a certain disease when its infection is 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 $0.1\%$ of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

$\int\frac{\text{x}^3}{\text{x}-2}\text{dx}$

Find $\frac{1}{2}\left(\mathrm{~A}+\mathrm{A}^{\prime}\right)$ and $\frac{1}{2}\left(\mathrm{~A}-\mathrm{A}^{\prime}\right)$ when $A=\left[\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right]$.

An urn contains 3 white, 4 red and 5 black balls. Two balls are drawn one by one without replacement. What is the probability that at least one ball is black?

If x = a (cos t + t sin t) and y = a (sin t – t cos t), find $\frac{{{d^2}y}}{{d{x^2}}}$