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

Question

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

✓

Answer

$\int\sin^{-1}\Big(\frac{2\tan\text{x}}{1+\tan^2\text{x}}\Big)\text{dx}$
$=\int\sin^{-1}(\sin2\text{x} )\text{dx}$ $\Big[\because\ \sin2\text{x}=\frac{2\tan\text{x}}{1+\tan^2\text{x}}\Big]$
$=\int2\text{ x dx}$
$=2\int\text{x dx}$
$=\frac{2\text{x}^2}{2}+\text{C}$
$=\text{x}^2+\text{C}$
$\therefore\ \int\sin^{-1}\Big(\frac{2\tan\text{x}}{1+\tan^2\text{x}}\Big)\text{dx}=\text{x}^2+\text{C}$

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

Similar questions

For the following differntial equations verify that the accompanying function is a solution:

Differential equation	Function
$\text{x}^3\frac{\text{d}{^2}\text{y}}{\text{dx}^2}=1$	$\text{y}=\text{ax}+\text{b}+\frac{1}{2\text{x}}$

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point '$c$' in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = 2x^2 - 3x + 1$ on $[1, 3]$

Evaluate the following definite integrals:
$\int_{0}^\limits{2}\frac{1}{\sqrt{3+2\text{x}-\text{x}^2}}\text{ dx}$

Write a value of $\int\text{e}^{3\log\text{x}}\text{x}^{4}\text{ dx}.$

Show that $y = \log \left( {1 + x} \right) - \frac{{2x}}{{2 + x}},x > - 1$ is an increasing function of x throughout its domain.

If $\text{A}=\text{diag}\begin{pmatrix}2&-5&9\end{pmatrix},\text{ B}=\text{diag}\begin{pmatrix}1&1&-4\end{pmatrix}$ and $\text{C}=\text{diag}\begin{pmatrix}-6&3&4\end{pmatrix},$ find.
$\text{B}+\text{C}-2\text{A}$

The volume of a cube is increasing at the rate of $9\ cm^3/ \sec.$ How fast is the surface area increasing when the length of an edge is $10\ cm?$

$\text{A}=\begin{bmatrix}\sin\alpha&\cos\alpha\\-\cos\alpha&\sin\alpha\end{bmatrix}$, then verify that A'A = I

Form the differential equation having $\text{y} = (\sin^{-1}\text{x})^{2} + \text{A}\cos^{-1} + \text{ B},$ where A and B are arbitrary constants, as its general solution.

If $\tan^{-1}\text{x}+\tan^{-1}\text{y}=\frac{\pi}{4},$ then write the value of x + y + xy.