Evaluate the following: ^ 2 _0 dx 1+m^2 ^2x dx - Vidyadip

Question

Evaluate the following:
$\int\limits^{\frac{\pi}{2}}_0\frac{\tan\text{x dx}}{1+\text{m}^2\tan^2\text{x}}\text{dx}$

✓

Answer

Let $\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\tan\text{x dx}}{1+\text{m}^2\tan^2\text{x}}\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_0\frac{\frac{\sin\text{x}}{\cos\text{x}}}{1+\text{m}^2\cdot\frac{\sin^2\text{x}}{\cos^2\text{x}}}\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_0\frac{\sin\text{x}\cos\text{x dx}}{1-\sin^2\text{x}+\text{m}^2\sin^2\text{x}}\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_0\frac{\sin\text{x}\cos\text{x}}{1+(\text{m}^2-1)\sin^2\text{x}}\text{dx}$
Put $\sin^2\text{x}=\text{t}$ $\Rightarrow\ 2\sin\text{x}\cos\text{x}\text{ dx}=\text{dt}$
When $\text{x}\rightarrow0,$ then $\text{t}\rightarrow0$ and $\text{x}\rightarrow\frac{\pi}{2},$ then $\text{t}\rightarrow1$
$\therefore\ \text{I}=\frac{1}{2}\int\limits^1_0\frac{\text{dt}}{1+(\text{m}^2-1)\text{t}}$
$=\frac{1}{2}\Big[\frac{1}{\text{m}^2-1}\log(1+(\text{m}^2-1)\text{t})\Big]^1_0$
$=\frac{1}{2(\text{m}^2-1)}\big[\log(1+(\text{m}^2-1))-\log1\big]$
$=\frac{\log\text{m}^2}{2(\text{m}^2-1)}=\frac{2\log|\text{m}|}{2(\text{m}^2-1)}=\frac{\log|\text{m}|}{\text{m}^2-1}$

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 "4 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

Find the particular solution of the differential equation $\frac{\text{dx}}{\text{dy}} + \text{x}\cot \text{y}=2\text{y} + \text{y}^{2} \cot \text{y},\text{ y}\neq0$ given that x = 0 when $\text{y}=\frac{\pi}{2}$

Find the maximum value of 2x³ - 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [-3, -1].

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}\cos(\text{x}-\text{y})=1$

For the following differntial equations verify that the accompanying function is a solution:

Differential equation	Function
$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{y}^2$	$\text{y}=\frac{\text{a}}{\text{x}+\text{a}}$

Prove that:
$\begin{vmatrix}\text{a}^2&\text{a}^2-(\text{b}-\text{c})^2&\text{bc}\\\text{b}^2&\text{b}^2-(\text{c}-\text{a})^2&\text{ca}\\\text{c}^2&\text{c}^2-(\text{a}-\text{b})^2&\text{ab}\end{vmatrix}$
$=(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{b})(\text{a}+\text{b}+\text{c})(\text{a}^2+\text{b}^2+\text{c}^2)$

If $\vec{\alpha}=3\hat{\text{i}}+4\hat{\text{j}}+5\hat{\text{k}}$ and $\vec{\beta}=2\hat{\text{i}}+\hat{\text{j}}-4\hat{\text{k}},$ then express $\vec{\beta}$ in the form of $\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2,$ where $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$.

Find the equation of the plane that contains the line of intersection of the planes $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})-4=0$ and $\vec{\text{r}}\cdot(2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}})+5=0$ and which is perpendicular to the plane $\vec{\text{r}}\cdot(5\hat{\text{i}}+3\hat{\text{j}}-6\hat{\text{k}})+8=0.$

A company manufactures two types of screws A and B. All the screws have to pass through a threading machine and a slotting machine. A box of Type A screws requires 2 minutes on the threading machine and 3 minutes on the slotting machine. A box of type B screws requires 8 minutes of threading on the threading machine and 2 minutes on the slotting machine. In a week, each machine is available for 60 hours.
On selling these screws, the company gets a profit of Rs 100 per box on type A screws and Rs 170 per box on type B screws.

Differentiate $\tan^{-1} \bigg(\frac{\sqrt{1 + x^{2} - 1}}{x}\bigg)$ w.r.t. $\sin^{-1} \frac{2x}{1 + x^{2}}, \text{ if } x \in (-1, 1)$

Differentiate the following functions with respect to x:
$\sqrt{\tan^{-1}\big(\frac{\text{x}}{2}\big)}$