Evaluate: ^ _ 0 x x . cosec x - Vidyadip

Question

Evaluate:
$\int\limits^{\pi}_{0}\frac{\text{x}\tan\text{x}}{\sec \text{x} . \text{cosec x}}$

✓

Answer

$\int\limits^{\pi}_{0}\frac{\text{x}\tan\text{x dx}}{\sec \text{x cosec x}}$
$\int\limits^{\pi}_{0}\text{x}\sin^{2}\text{x dx} $
$\text{Let I} = \int\limits^{\pi}_{0}\text{x}\sin^{2}\text{x dx}$
$= \int\limits^\pi_0(\pi -\text{x})\sin^{2}(\pi - \text{x) dx}$
$= \int\limits^{\pi}_{0}(\pi - \text{x)}\sin^{2}\text{x dx}$
$\text{2 I}= \pi\int\limits\sin^{2}\text{x dx} = \pi\int\limits^{\pi}_{0}\frac{1 - \cos\text{2 x}}{2}\text{dx}$
$= \frac{\pi}{2} \Bigg[\text{x} - \frac{\sin\text{2 x}}{2}\Bigg]^{\pi}_{0}$
$= \frac{\pi^{2}}{2}$
$\text{I} = \frac{\pi^{2}}{4}$

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 APPLICATION OF DERIVATIVES→APPLICATION OF DERIVATIVES — 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 vector and Cartesian equations of the plane that passes through the point (5, 2, -4) and is perpendicular to the line with direction ratios 2, 3, -1.

Find the probability distribution of the number of doublets in three throws of a pair of dice and find its mean.

Find the integrals of the functions in Exercises:
$\frac{1}{\cos(\text{x}-\text{a})\cos(\text{x}-\text{b})}$

If $\text{y}=\text{a}\{\text{x}+\sqrt{\text{x}^2+1}\}^\text{n}+\text{b}\{\text{x}-\sqrt{\text{x}^2+1}\}^{-\text{n},}$ prove that $(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}-\text{n}^2\text{y}=0.$

Evaluate the following integrals:
$\int\frac{\log\big(1+\frac{1}{\text{x}}\big)}{\text{x}(1+\text{x})}\text{dx}$

Find the angle of intersection of the curves $y^2=4 a x$ and $x^2=4 b y$.

If $y = 500e^{7x} + 600e^{-7x}$ show that $\frac{{{d^2}y}}{{d{x^2}}} = 49y$.

A box manufacturer makes large and small boxes from a large piece of cardboard. The large boxes require 4 sq. metre per box while the small boxes require 3 sq. metre per box. The manufacturer is required to make at least three large boxes and at least twice as many small boxes as large boxes. If 60 sq. metre of cardboard is in stock, and if the profits on the large and small boxes are Rs. 3 and Rs. 2 per box, how many of each should be made in order to maximize the total profit?

If $\text{y}=(\sin^{-1}\text{x})^2,$ prove that $(1-\text{x}^2)\text{y}_2-\text{xy}_1-2=0$

Evaluate the following integrals as limit of sum:
$\int\limits^3_1(3\text{x}-2)\text{dx}$