Evaluate the following integrals: _ 0 ^ 4 ^ 3 x 1+ 2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{4}}\frac{\tan^{3}\text{x}}{1+\cos2\text{x}}\text{ dx}$

✓

Answer

Let $\text{I}=\int_{0}^\limits{\frac{\pi}{4}}\frac{\tan^{3}\text{x}}{1+\cos2\text{x}}\text{ dx}$ Then,
$\text{I}=\int_{0}^\limits{\frac{\pi}{4}}\frac{\tan^{3}\text{x}}{1+\cos2\text{x}}\text{ dx}$
$\Rightarrow\text{I}=\frac{1}{2}\int_{0}^\limits{\frac{\pi}{4}}\tan^{3}\text{x}\sec^2\text{x}\text{ dx}$
Let $\tan\text{x}=\text{t}$ Then, $\sec^2\text{x dx}=\text{dt}$
When $\text{x}=0,\text{t}=0$ and $\text{x}=\frac{\pi}{4},\text{t}=1$
$\therefore\ \text{I}=\frac{1}{2}\int_{0}^\limits{1}\text{t}^3\text{ d}t$
$\Rightarrow\text{I}=\frac{1}{2}\Big[\frac{\text{t}^4}{4}\Big]^1_0$
$\Rightarrow\text{I}=\frac{1}{2}\Big(\frac{1}{4}-0\Big)$
$\Rightarrow\text{I}=\frac{1}{8}$

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 "5 Marks Questions" from Definite Integrals→Definite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If $\vec{\text{a}},\vec{\text{ b}}$ and $\vec{\text{c}}$ determine the vertices of a triangle, show that $\frac{1}{2}[\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{a}}+\vec{\text{a}}\times\vec{\text{b}}]$ gives the vector area of the triangle. Hence, deduce the condition that the three points $\vec{\text{a}},\vec{\text{ b}}$ and $\vec{\text{c}}$ are collinear. Also, find the unit vector normal to the plane of the triangle.

$\begin{vmatrix}\text{a}+\text{b}+\text{c}&-\text{c}&-\text{b}\\-\text{c}&\text{a}+\text{b}+\text{c}&-\text{a}\\-\text{b}&-\text{a}&\text{a}+\text{b}+\text{c}\end{vmatrix}$
$=2(\text{a}+\text{b})(\text{b}+\text{c})(\text{c}+\text{a})$

A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs. 7 profit and that of B at a profit of Rs. 4. Find the production level per day for maximum profit graphically.

Find the equation of the plane passing through the line of intersection of the planes $\vec{\text{r}}.\Big(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\Big)=1\ \text{and}\ \vec{\text{r}}.\Big(2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}\Big)+4=0$ and parallel to x-axis.

Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix}2 & 5 \\ 1 & 3 \end{bmatrix}$

Using matrices, solve the following system of linear equations: $x – y + 2z = 7., 3x + 4y - 5z = –5. ,2x – y + 3z = 12.$

If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are non-zero, non-coplanar vectors, prove that the vector is coplanar:
$\vec{\text{a}}-2\vec{\text{b}}+3\vec{\text{c}},\ -3\vec{\text{b}}+5\vec{\text{c}}$ and $-2\vec{\text{a}}+3\vec{\text{b}}-4\vec{\text{c}}$

Evaluate the following integrals:$\int\frac{\sin2\text{x}}{\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}}\text{ dx}$

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\cos\text{x}+\sin\text{x}}{\cos\text{x}-\sin\text{x}}\Big), \frac{\pi}{4}<\text{x}<\frac{\pi}{4}$

Draw a rough sketch of the region $\{(x, y) : y^2 < 3x, 3x^2 + < 16\}$ and find the area by the region using mwthod of integration.