Integrate the functions in Exercises: e^ ^ -1 x 1+x^2 - Vidyadip

Question

Integrate the functions in Exercises:
$\frac{\text{e}^{\tan^{-1}\text {x}}}{1+\text{x}^2}$

✓

Answer

$\text{Let I}=\int\frac{\text{e}^{\tan^{-1}\text{x}}}{1+\text{x}^2}\text{ dx} \ \ \ \ \ \ \ ...\text{(i)} $
$\text{Putting }\tan^{-1}\text{x}=\text{t}\ \ \ \Rightarrow \ \ \ \frac{1}{1+\text{x}^2}=\frac{\text{dt}}{\text{dx}}\ \ \ \ \Rightarrow \ \ \ \ \frac{\text{dx}}{1+\text{x}^2}=\text{ dt} $
$\therefore \ \ \ \ $From eq. (i), $\text{I}=\int\text{e}^{\text{t}}\text{ dt}=\text{e}^{\text{t}}+\text{c}=\text{e}^{{\tan}^{-1}\text{x}}+\text{c}$

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 "1 Marks" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The area of the region bounded by the circle x² + y² = 1 is:

$2\pi\text{ sq.}\text{ units}$
$\pi\text{ sq.}\text{ units}$
$3\pi\text{ sq.}\text{ units}$
$4\pi\text{ sq.}\text{ units}$

If the matrix A is both symmetric and skew symmetric, then

For a 2 × 2 matrix, A = [a_ij], whose elements are given by a_ij= $\frac{\text{i}}{\text{j}}$, write the value of a₁₂.

Find the principal value of the following:
$\sin^{-1}\Big(-\frac{\sqrt3}{2}\Big)$

Integrate the function x cos^-1 x

Integrate the function ${e^x}\left( {\frac{1}{x} - \frac{1}{{{x^2}}}} \right)$

If f(x) = x + 1, find $\frac{\text{d}}{\text{dx}}(\text{fof})(\text{x}).$

Evaluate $\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{4} x}{\sin ^{4} x+\cos ^{4} x} d x$

Write a unit vector in the direction of the sum of the vectors $\vec{\text{a}}=2\hat{\text{i}}+2\hat{\text{j}}-5\hat{\text{k}}\ \text{and}\ \vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}-7\hat{\text{k}}$

Find the order and the degree of the differential equation $\text{x}^2\frac{\text{d}^2\text{y}}{\text{dx}^2}=\bigg\{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\bigg\}^4.$