Evalute the following integrals: 1- 2x 1+ 2x dx - Vidyadip

Question

Evalute the following integrals:
$\int\sqrt{\frac{1-\sin2\text{x}}{1+\sin2\text{x}}}\text{dx}$

✓

Answer

Let $\text{I}=\int\sqrt{\frac{1-\sin2\text{x}}{1+\sin2\text{x}}}\text{dx}$ then,
$=\int\sqrt{\frac{1-\cos\Big(\frac{\pi}{2}-2\text{x}\Big)}{1+\cos\Big(\frac{\pi}{2}-2\text{x}\Big)}}\text{dx}$
$=\int\sqrt{\frac{2\sin^2\Big(\frac{\pi}{4}-\text{x}\Big)}{2\cos^2\Big(\frac{\pi}{4}-\text{x}\Big)}}\text{dx}$
$=\int\sqrt{\tan^2\Big(\frac{\pi}{4}-\text{x}\Big)}\text{dx}$
$=\int\tan\Big(\frac{\pi}{4}-\text{x}\Big)\text{dx}$
$=\log\Big|\cos\Big(\frac{\pi}{4}-\text{x}\Big)\Big|+\text{C}$

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

$\text{If}\ \ \vec{\text{a}}=\vec{\text{b}}+\vec{\text{c}},$ then is it true that $\big|\vec{\text{a}}\big|=\big|\vec{\text{b}}\big|+\big|\vec{\text{c}}\big|?$ Justify your answer.

Using the property of determinants and without expanding, prove that:
$\begin{vmatrix}-a^{2}&ab&ac\\ba&-b^{2}&bc\\ca&cb&-c^{2}\end{vmatrix}=4a^2b^2c^2$

Examine the consistency of the system of equations:

3x - y - 2z = 2

Find the projection of $\vec{\text{b}}+\vec{\text{c}}$ on $\vec{\text{a}},$ where $\vec{\text{a}}=2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}$ and $\vec{\text{c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}.$

Solve $\text{y}\text{dx}-\text{x}\text{dy}=\text{x}^2\text{y}\text{dx}.$

Evaluate the following intregals:
$\int\frac{\text{x}}{\sqrt{\text{x}^2+\text{x}+1}}\text{dx}$

If a unit vector $\vec{\text{a}}$ makes an angle $\frac{\pi}{3}$ with $\hat{\text{i}},\frac{\pi}{4}$ with $\hat{\text{j}}$ and an acute angle $\theta$ with $\hat{\text{k}}$, and ,then find the value of $\theta$.

Write the following in the simplest form:
$\tan^{-1}\Big\{\sqrt{1+\text{x}^2}-\text{x}\Big\},\text{x}\in\text{R}$

In each of the verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

$\text{y} – \text{cos}\ \text{y} = \text{x} \ :\ (\text{y} \ \text{sin} \ \text{y} + \text{cos} \ \text{y} + \text{x}) \text{y}' = \text{y}$

If $\big[3\vec{\text{a}}+7\vec{\text{b}}\vec{\text{c}}\vec{\text{d}}\big]=\lambda\big[\vec{\text{a}}\vec{\text{c}}\vec{\text{d}}\big]+\mu\big[\vec{\text{b}}\vec{\text{c}}\vec{\text{d}}\big],$ then find the value of $\lambda+\mu.$