Evaluate: ^ _0x x sec x. cosec x dx. - Vidyadip

Question

Evaluate: $\int\limits^\pi_0\frac{x\tan x}{\text{sec }x.\text{ cosec }x}\text{ d}x.$

✓

Answer

$\text{Let I}=\int\limits^\pi_0\frac{\text{x tan x}}{\text{sec x cosec x}}\text{ dx}$
$\therefore\text{ I}=\int\limits^\pi_0\frac{(\pi-\text{x})\tan(\pi-\text{x})}{ \sec (\pi-\text{x}) \text{ cosec} (\pi-\text{x})}\text{ dx}$
$\Rightarrow\text{ I}=\int\limits^\pi_0\frac{(\pi-\text{x})\tan\text{x}}{ \sec\text{x}.\text{cosec}\text{ x}}\text{ dx}$
$\text{Adding we ge t, 2I}=\pi\int\limits^\pi_0\frac{\tan\text{x}}{ \sec\text{x}.\text{cosec}\text{ x}}\text{ dx}$
$\text{2I}=\pi\int\limits^\pi_0\sin^2\text{x}\text{ dx}$
$=2\pi\int\limits^{\pi/2}_0\frac{1-\cos2\text{x}}{2}\text{ dx}$
$=\pi\bigg[\Big(\text{x}-\frac{\sin2\text{x}}{2}\Big)\bigg]^{\pi/2}_0$
$=\pi.\frac{\pi}{2}=\frac{\pi^2}{2}\ \ \therefore\ \ \text{I}=\frac{\pi^2}{4}$

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 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

Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\cos^4\text{x}\text{ dx}$

If $\text{x}=\cot\text{t and y}=\sin\text{t},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{1}{\sqrt{3}}\text{ at t}=\frac{2\pi}{3}$

Differentiate $\frac{\text{x}}{\sin\text{x}}$ w.r.t. $\sin\text{x}.$

Find the equation of the normal to the curve $x^2 + 2y^2 - 4x - 6y + 8 = 0$ at the point whose abscissa is 2.

If x and y are connected parametrically by the equation $x=\frac{\sin ^{3} t}{\sqrt{\cos 2 t}}, y=\frac{\cos ^{3} t}{\sqrt{\cos 2 t}}$, without eliminating the parameter, Find $\frac{d y}{d x}$

Evaluate the following integrals:
$\int\limits^1_{-1}|\text{x}\cos\pi\text{x}|\text{dx}$

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{4\text{x}}{1-4\text{x}^2}\Big),-\frac{1}{2}<\text{x}<\frac{1}{2}$

Evaluate the following definite integrals:
$\int_{\frac{\pi}{3}}^\limits{\frac{\pi}{4}}(\tan\text{x}+\cot\text{x})^2\text{ dx}$

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
$4x + 8y + z - 8 = 0$ and $y + z - 4 = 0$

Find the adjoint of the matrix $\text{A}=\begin{bmatrix} -1 & -2 & -2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{bmatrix}$ and hence show that $A (adj\ A) = |A|I_3.$