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

Question

Evaluate the following integrals:$\int\text{e}^{\text{x}}.\frac{\sqrt{1-\text{x}^2}\sin^{-1}\text{x}+1}{\sqrt{1-\text{x}^2}}\text{dx}$

✓

Answer

Let $\text{I}=\int\text{e}^{\text{x}}\bigg[\frac{\sqrt{1-\text{x}^2}\sin^{-1}\text{x}+1}{\sqrt{1-\text{x}^2}}\bigg]\text{dx}$
$=\int\text{e}^{\text{x}}\Big[\sin^{-1}\text{x}+\frac{1}{\sqrt{1-\text{x}^2}}\Big]\text{dx}$
Here, $\text{f(x)}=\sin^{-1}\text{x}$
$\Rightarrow\text{f}'\text{(x)}=\frac{1}{\sqrt{1-\text{x}^2}}$
Put $\text{e}^{\text{x}}\text{f(x)}=\text{t}$
$\Rightarrow\text{e}^{\text{x}}\sin^{-1}\text{x}=\text{t}$
Diff both sides w.r.t x
$\Big(\text{e}^{\text{x}}\sin^{-1}\text{x}+\text{e}^{\text{x}}\times\frac{1}{\sqrt{1-\text{x}^2}}\Big)\text{dx = dt}$
$\because\text{I}=\int\text{dt}$
$=\text{t + C}$
$=\text{e}^{\text{x}}\sin^{-1}\text{x + C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $\text{A}=\begin{bmatrix}2&3\\-1&0\end{bmatrix},$ show that $A^2 - 2A + 3I_2 = 0.$

$\triangle O A B$ is formed by lines $x^2-4 x y+y^2=0$ and the line $2 x+3 y-1=0$. Find the equation of the median of the triangle drawn from $\mathrm{O}$.

A couple has two children. Find the probability that both the children are,

Males, if it is known that at least one of the children is male.
Females, if it is known that the elder child is a female.

Differentiate $\tan^{-1}\Big(\frac{1+\text{ax}}{1-\text{ax}}\Big)$ with respect to $\sqrt{1+\text{a}^2\text{x}^2}$

Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis twice abscissa of the pont of contact.

Differentiate the following functions with respect to x:
$\text{e}^{\sin\text{x}}+(\tan\text{x})^\text{x}$

If a, b and c are all non-zero and $\begin{vmatrix}1+\text{a}&1&1\\1&1+\text{b}&1\\1&1&1+\text{c} \end{vmatrix}=0,$ then prove that $\frac{1}{\text{a}}+\frac{1}{\text{b}}+\frac{1}{\text{c}}+1=0.$

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{e}^{3\text{x}}\sin4\text{x}\times2^\text{x}$

If $\text{x}=\text{a}\sec\theta,\text{y}=b\tan\theta$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\frac{\text{b}^4}{\text{a}^2\text{y}^3}$

If $\sqrt{1-\text{x}^2}+\sqrt{1-\text{y}^2}=\text{a}(\text{x}-\text{y}),$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sqrt{1-\text{y}^2}}{1-\text{x}^2}$