Evaluate: _0^ 2 (2 x - 2x) dx - Vidyadip

Question

$\text{Evaluate:} \int\limits_0^\frac{\pi}{2} (2\log \sin \text{x} - \log \sin 2\text{x}) \text{dx}$

✓

Answer

$\text{I} = \int\limits_0^\frac{\pi}{2} (2\log \sin \text{x} - \log \sin 2\text{x}) \text{dx} = \int\limits_0^\frac{\pi}{2}\log\bigg(\frac{\sin^{2}{x}}{2\sin x \cos x}\bigg)\text{dx}$
$= \int\limits_0^\frac{\pi}{2} \log \tan \text{x dx} - \int\limits_0^\frac{\pi}{2} \log 2 .\text{dx} = \text{I}_{1} - \text{I}_{2}$
$\text{I}_{1} = \int\limits_0^\frac{\pi}{2} \log \tan \text{x dx} = \int\limits_0^\frac{\pi}{2} \log \tan\bigg(\frac{\pi}{2} - \text{x}\bigg) \text{dx} = \int\limits_0^\frac{\pi}{2} \log \cot\text{x dx}$
$\Rightarrow \text{2 I}_{1} = 0 \Rightarrow \text{I}_{1} = 0$
$\text{I}_{2} = \log 2. \bigg[\text{x}\bigg]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} . \log 2$
$\therefore \text{I} = - \frac{\pi}{2} . \log 2$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

A swimming pool is to be drained for cleaning. If $L$ represents the number of litres of water in the pool $t$ seconds after the pool has been plugged off to drain and $L = 200(10 - t)^2.$ How fast is the water running out at the end of $5$ seconds$?$ What is the average rate at which the water flows out during the first $5$ seconds$?$

Let $R$ be relation defined on the set of natural number $N$ as follows: $R=\{(x, y): x \in N, y \in N, 2 x+y=41\}$. Find the domain and range of the relation $R$. Also verify whether $R$ is reflexive, symmetric and transitive.

Find the area enclosed by the curve $y = -x^2$ and the strainght line $x + y + 2 = 0.$

Differentiate the following functions with respect to x:
$\log\sqrt{\frac{\text{x}-1}{\text{x}+1}}$

Prove that $\begin{vmatrix}\text{bc}-\text{a}^2&\text{ca}-\text{b}^2&\text{ab}-\text{c}^2\\\text{ca}-\text{b}^2&\text{ab}-\text{c}^2&\text{bc}-\text{a}^2\\\text{ab}-\text{c}^2&\text{bc}-\text{a}^2&\text{ca}-\text{b}^2\end{vmatrix}$ is divisible by $(a + b + c)$ and find the quotient.

Find the equation of a plane passing through the point $\text{P (6, 5, 9)}$ and parallel to the plane determined by the points $\text{A (3, –1, 2), B (5, 2, 4)}$ and $\text{ C (–1, –1, 6)}$ . Also find the distance of this plane from the point A.

A fair coin is tossed four times. Let X denote the longest string of heads accuring. Find the probability distribution mean and variance of X.

Solve the following systems of linear equations by cramer's rule:

2x + ay = 2, $\text{a}\neq0$

The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.

Solve the following differential equation:
$\Big(1+\text{e}^{\frac{\text{x}}{\text{y}}}\Big)\text{dx}+\text{e}^{\frac{\text{x}}{\text{y}}}\Big(1-\frac{\text{x}}{\text{y}}\Big)\text{dy}=0$