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

Question

Evaluate the following integrals:
$\int\tan^{-1}\sqrt{\frac{1-\text{x}}{1+\text{x}}}\text{dx}$

✓

Answer

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

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 "4 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

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes.

Find the intervals in which the following functions are increasing or decreasing.
f(x) = x²+ 2x - 5

Maximize Z = 50x + 30y
Subject to
$2\text{x}+\text{y}\leq18$
$3\text{x}+2\text{y}\leq34$
$\text{x},\text{y}\geq0$

A manufacturer of Furniture makes two products : chairs and tables. processing of these products is done on two machines A and B. A chair requires 2 hrs on machine A and 6 hrs on machine B. A table requires 4 hrs on machine A and 2 hrs on machine B. There are 16 hrs of time per day available on machine A and 30 hrs on machine B. Profit gained by the manufacturer from a chair and a table is Rs. 3 and Rs. 5 respectively. Find with the help of graph what should be the daily production of each of the two products so as to maximize his profit.

Differentiate the following functions with respect to x:
$\sin^{-1}\Big\{\frac{\sin\text{x}+\cos\text{x}}{\sqrt{2}}\Big\},-\frac{3\pi}{4}<\text{x}<\frac{\pi}{4}$

Solve the following system of equations by matrix method:
2x + y + z = 2
x + 3y − z = 5
3x + y − 2z = 6

In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}5,&\text{if }\text{ x}\leq2\\\text{ax}+\text{b},&\text{if }2<\text{x}<10\\21,&\text{if }\text{ x}\geq10\end{cases}$

Find the point on the curve y = x² - 2x + 3, where the tangent is parallel to x-axis.

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

$\text{If y} = \tan^{-1} \bigg(\frac{\sqrt{1 + x^{2}}+{\sqrt{1 - x^{2}}}}{\sqrt{1 + x^{2}} - {\sqrt{1 - x^{2}}}}\bigg), x^{2}\leq 1, \text{then find} \frac{dy}{dx}.$