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

Question

Evaluate the following integrals:

$\int\frac{\sec^2\text{x}}{1-\tan^2\text{x}}\text{dx}$

✓

Answer

$\int\frac{\sec^2\text{x dx}}{1-\tan^2\text{x}}$
Let $\tan\text{x = t}$
$\Rightarrow\sec^2\text{x dx = dt}$
Now, $\int\frac{\sec^2\text{x dx}}{1-\tan^2\text{x}}$
$=\int\frac{\text{dt}}{1-\text{t}^2}$
$=\frac{1}{2}\log\bigg|\frac{1+\text{t}}{1-\text{t}}\bigg|+\text{C}$
$=\frac{1}{2}\log\bigg|\frac{1+\tan\text{x}}{1-\tan\text{x}}\bigg|+\text{C}$

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" 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

Find the matrix A such that
$\text{A}=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\\11&10&9\end{bmatrix}$

Find the area of the parallelogram whose diagonals are:
$3\hat{\text{i}}+4\hat{\text{j}}$ and $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
2x + y + 3z - 2 and x - 2y + 5 = 0

Show that the line through the points (1, -1, 2) and (3, 4, -2) is perpendicular to the through the points (0, 3, 2) and (3, 5, 6).

Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{y}=\text{x}^3-\text{x}\ \text{at}\ \text{x}=2$

If A is a 3 × 3 invertible matrix, then what will be the value of k if det (A^-1) = (det A)^k.

For any vector $\vec{a}$ prove that $|\vec{a} \times \hat{i}|^2+|\vec{a} \times \hat{j}|^2+|\vec{a} \times \hat{k}|^2=2|a|^2$

Evaluate the following integrals:

$\int\frac{\text{x}+5}{3\text{x}^2+13\text{x}-10}\text{ dx}$

verify that $\text{y}=\text{cx}+2\text{c}^2$ is a solution of the differential equation $2\Big(\frac{\text{d}\text{y}}{\text{dx}}\Big)^2-\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=0$

Solve the following equation:$\cos(\tan^{-1}\text{x}) = \sin\bigg(\cot^{-1}\frac{3}{4}\bigg).$