Evaluate: _0^ ^2 x d x - Vidyadip

Question

Evaluate: $\int_0^\pi \cos ^2 x d x$

✓

Answer

$\int_0^\pi \cos ^2 x d x=\int_0^\pi\left(\frac{1+\cos 2 x}{2}\right) d x$
$=\frac{1}{2}\left[\int_0^\pi d x+\int_0^\pi \cos 2 x d x\right]$
$=\frac{1}{2}\left[[x]_0^\pi+\left[\frac{\sin 2 x}{2}\right]_0^\pi\right]$
$=\frac{1}{2}\left[(\pi-0)+\frac{1}{2}(\sin 2 \pi-\sin 0)\right]$
$=\frac{1}{2}\left[\pi+\frac{1}{2}(0-0)\right]$
$=\frac{\pi}{2}$

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 "Solve the Following Question.(2 Marks)" from Question Bank [2022]→Question Bank [2022] — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $y=\cos ^{-1}\left(1-2 \sin ^2 x\right)$, find $\frac{d y}{d x}$

A r.v. X ~ B(n, p). If the values of mean and variance of X are 18 and 12 respectively, then find total number of positive values of X.

If $|\vec{\text{a}}|=10,\big|\vec{\text{b}}\big|=2$ and $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=16,$ find $\vec{\text{a}}.\vec{\text{b}}.$

Find the angle between the line $\bar{r}=(\hat{i}+2 \hat{j}+\hat{k})+\lambda(\hat{i}+\hat{j}+\hat{k})$ and the plane $\bar{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=8$

For the set A = {1, 2, 3}, define a relation R on the set A as follows:
R = {(1, 1), (2, 2), (3, 3), (1, 3)}
Write the ordered pairs to be added to R to make the smallest equivalence relation.

Evaluate the following:
$\cot\Big(\cot^{-1}\frac{3}{5}\Big)$

Let $f: R \rightarrow R^{+}$be defined by $f(x)=a x, a>0$ and $a \neq 1$. Write $f^{-1}(x)$.

Determine the order and degree of each of the following differential equations:

$\left(\frac{d^3 y}{d x^3}\right)^{\frac{1}{2}} \cdot\left(\frac{d y}{d x}\right)^{\frac{1}{3}}=20$

Integrate the following with respect to the respective variable:

$\frac{x^7}{x+1}$

Determine whether the following statement patterns are tautologies contradictions or contingencies : (p → q) ∧ (p ∧ ~q)