Evaluate the following integrals: ^ 4 _ - 4 ^2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\sin^2\text{x dx}$

✓

Answer

Let $\text{I}=\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\sin^2\text{x dx}$
Here $\text{f(x)}=\sin^2\text{x}$
$\text{f}(-\text{x})=\sin^2(-\text{x})=\sin^2\text{x}=\text{f(x)}$
Hence $\sin^2\text{x}$ is an even function
Therefore,
$\text{I}=2\int\limits^{\frac{\pi}{4}}_{0}\sin^2\text{x dx}$
$=2\int\limits^{\frac{\pi}{4}}_{0}\Big(\frac{1-\cos2\text{x}}{2}\Big)\text{dx}$
$=\int\limits^{\frac{\pi}{4}}_{0}(1-\cos2\text{x})\text{dx}$
$=\Big[\text{x}-\frac{\sin^2\text{x}}{2}\Big]^{\frac{\pi}{4}}_0$
$=\frac{\pi}{4}-\frac{1}{2}$

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 Definite Integrals→Definite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the area enclosed by the parabolas $y= 4x - x^2$ and $y = x^2 - x.$

Find the absolute maximum and minimum values of the function f given by $f(x) = \cos^2 x + \sin x, x \in [0, \pi]$

Evaluate the following intregals:
$\int\frac{5\text{x}^2+20\text{x}+6}{\text{x}^2+2\text{x}^2+\text{x}}\ \text{dx}$

Using properties of determinants show that $\begin{vmatrix} 1 & 1 & \text{1 + x} \\ 1 & \text{1 + y} & 1 \\ \text{1 + z} & 1 & 1 \end{vmatrix} = \text{xyz + yz + zx + xy}.$

In order to supplement daily diet, a person wishes to take $X$ and $Y$ tablets. The contents $($in milligrams per tablet$)$ of iron, calcium and vitamins in $X$ and $Y$ are given as below:

Tablets	Iron	Calcium	Vitamin
$X$	$6$	$3$	$2$
$Y$	$2$	$3$	$4$

The person needs to supplement at least $18$ milligrams of iron, $21$ milligrams of calcium and $16$ milligrams of vitamins. The price of each tablet of $X$ and $Y$ is $₹ 2$ and $₹1$ respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an $LPP$ and solve graphically.

Find the value of a and b so that the function f given by $\text{f(x)}=\begin{cases}1,&\text{if }\text{ x}\leq3\\\text{ax}+\text{b},&\text{if }3<\text{x}<5\\7,&\text{if }\text{ x}\geq5\end{cases}$ is continuous x = 3 and x = 5.

Solve the following system of equations by matrix method:
$3x + 4y + 2z = 8$
$2y - 3z = 3$
$x - 2y + 6z = -2$

Evaluate the following integrals:
$\int\text{e}^{\text{x}}(\log\text{x}+\frac{1}{\text{x}^2})\text{dx}$

Five forces $\overrightarrow{\text{AB}},\ \overrightarrow{\text{AC}},\ \overrightarrow{\text{AD}},\ \overrightarrow{\text{AE}}\text{ and }\overrightarrow{\text{AF}}$ act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is:$6\ \overrightarrow{\text{AO}}$, where o is the center of hexagon.

Evaluate the following intergrals:
$\int\text{e}^\text{ax}\cos\text{bx dx}$