1 ^2x(1- )^2 dx - Vidyadip

Question

$\int\frac{1}{\cos^2\text{x}(1-\tan\text{x})^2}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{1}{\cos^2\text{x}(1-\tan\text{x})^2}\text{dx}$
$=\int\frac{\sec^2\text{x}}{(1-\tan\text{x})^2}\text{dx}$
$=\int\frac{\sec^2\text{x dx}}{(1-\tan\text{x})^2}$
$-\sec^2\text{x dx}=\text{dt}$
$\Rightarrow\sec^2\text{x dx}=-\text{dt}$
$\therefore\text{I}=\int\frac{-\text{dt}}{\text{t}^2}$
$=-\int\text{t}^{-2}\text{dt}$
$=-\bigg[\frac{\text{t}^{-2+1}}{-2+1}\bigg]+\text{c}$
$=\frac{1}{\text{t}}+\text{c}$
$=\frac{1}{1-\tan\text{x}}+\text{c}$

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 "3 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

Integrate the function in Exercise:

$\sqrt{\frac{1-\sqrt{\text{x}}}{1+\sqrt{\text{x}}}}$

Prove the following results:
$\frac{9\pi}{4}-\frac{9}{4}\sin^{-1}\frac{1}{3}=\frac{9}{4}\sin^{-1}\frac{2\sqrt2}{3}$

$\text{Prove that:} \sin^{-1} \bigg(\frac{4}{5}\bigg) + \sin^{-1} \bigg(\frac{5}{13}\bigg) + \sin^{-1} \bigg(\frac{16}{65}\bigg) = \frac{\pi}{2}$

Evalute the following integrals:
$\int\frac{\sin2\text{x}}{\sin\Big(\text{x}-\frac{\pi}{6}\Big)\sin\Big(\text{x}+\frac{\pi}{6}\Big)}\text{dx}$

Evaluate the following integrals:

$\int\frac{\text{x}+\sin\text{x}}{1+\cos\text{x}}\text{dx}$

Evaluate the integral $\int_{0}^{2} x \sqrt{x+2}$ (Put x + 2 =t²) using substitution.

Find the direction cosines of the vector joining the points A(1, 2, -3) and B(-1, -2, 1) directed from A to B.

Let $\text{A}=\begin{bmatrix}1&-1&0\\2&1&3\\1&2&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&2&3\\2&1&3\\0&1&1\end{bmatrix},$ Find A^T, B^T and verify that.

$(2\text{A})^\text{T}=2\text{A}^\text{T}$

Evaluate the following integrals:

$\int\tan^{-1}\Big(\frac{3\text{x}-\text{x}^3}{1-3\text{x}^2}\Big)\text{dx}$

If $|\vec{\text{a}}|=2,\big|\vec{\text{b}}\big|=5$ and $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=8,$ find $\vec{\text{a}}.\vec{\text{b}}.$