Evaluate: ^ _ 0 x dx a^ 2 ^ 2 x + b^ 2 ^ 2 x . - Vidyadip

Question

Evaluate: $\int\limits^{\pi}_{0}\frac{\text{x dx}}{\text{a}^{2}\cos^{2}\text{x + b}^{2}\sin^{2}\text{x}}.$

✓

Answer

$\text{I}=\int\limits^{\pi}_{0}\frac{\text{x dx}}{\text{a}^{2}\cos^{2}\text{x + b}^{2}\sin^{2}\text{x}}=\int\limits_{0}^{\pi}\frac{(\pi-\text{x) dx}}{\text{a}^{2}\cos^{2}\text{x + b}^{2}\sin^{2}\text{x}}$
$\therefore2\text{I}=\pi\int\limits^{\pi}_{0}\frac{\text{dx}}{\text{a}^{2}\cos^{2}\text{x + b}^{2}\sin^{2}\text{x}}=2\pi\int\limits_{0}^{\pi/2}\frac{\sec^{2}\text{x}}{\text{a}^{2}+\text{b}^{2}\text{ }\tan^{2}\text{x}}\text{dx}$
$ =2\pi\int\limits^{\infty}_{0}\frac{\text{dt}}{\text{a}^{2}+\text{b}^{2}\text{ t}^{2}}=\frac{2\pi}{\text{ab}}\Bigg[\tan^{-1}\frac{\text{bt}}{\text{a}}\Bigg]^{\infty}_{0}\text{where tan x = t}$
$\text{2I}=\frac{2\pi}{\text{ab}}\cdot\frac{\pi}{2}=\frac{\pi^{2}}{\text{ab}}$
$\Rightarrow\text{I}=\frac{\pi^{2}}{\text{2ab}}.$

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

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},$ are three vectors such that $|\overrightarrow{a}|=5,|\overrightarrow{b}|=12\text{ and }|\overrightarrow{c}|=13,$ and $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{o},$ find the value of $\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}.$

Evaluate the following intregals:
$\int\frac{\text{x}}{\sqrt{\text{x}^2+6\text{x}+10}}\ \text{dx}$

Evalute the following integrals:
$\int\tan2\text{x}\tan3\text{x}\tan5\text{x dx}$

If $A =\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]$, then, prove that $A^3-6 A^2+7 A+2 I=O$ and hence find $A^{-1}$.

Let $\text{f(x)}=\begin{cases}\frac{1-\sin^3\text{x}}{3\cos^2\text{x}},&\text{if }\text{ x}<\frac{\pi}{2}\\\text{a},&\text{if }\text{ x}=\frac{\pi}{2}\\\frac{\text{b}(1-\sin\text{x})}{(\pi-2\text{x})}^2,&\text{x}>\frac{\pi}{2}\end{cases}$ if f(x) is continuous at $\text{x}=\frac{\pi}{2},$ find a and b.

Find the angle of intersecting of the following curves:
$\text{y}=\text{x}^2\text{ and }\text{x}^2+\text{y}^2=20$

Integrate the function in Exercise:
$\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$

Given $\text{A}=\begin{bmatrix}5 & 0 & 4 \\2 & 3 & 2 \\ 1 & 2 & 1\end{bmatrix},\text{B}^{-1}=\begin{bmatrix}1 & 3 & 3 \\1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}.$ Compute (AB)^-1.

$\text{if y } = \sqrt{\frac{(x - 3)(x^{2} + 4)}{3x^{2} + 4x + 5}}, \text{find} \frac{dy}{dx}$

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\sin\text{x}-\sin2\text{x}\text{ on }[0,\pi]$