Evaluate : _0^ x ^2 x d x - Vidyadip

Question

Evaluate : $\int_0^\pi x \cdot \sin ^2 x \cdot d x$

✓

Answer

$
\begin{aligned}
& \text { Consider, I } \quad=\int_0^\pi x \cdot \sin ^2 x \cdot d x \ldots \\
& \mathrm{I}=\int_0^\pi(\pi-x) \cdot[\sin (\pi-x)]^2 x \cdot d x \\
& \mathrm{I}=\int_0^\pi(\pi-x) \cdot \sin ^2 x \cdot d x \\
& \mathrm{I}=\int_0^\pi \pi \cdot \sin ^2 x \cdot d x-\int_0^\pi x \cdot \sin ^2 x \cdot d x \\
& \mathrm{I}=\pi \cdot \int_0^\pi \frac{1}{2}(1-\cos 2 x) \cdot d x-\mathrm{I} \ldots \text { by }(\mathrm{i}) \\
& \mathrm{I}+\mathrm{I}=\frac{\pi}{2} \int_0^\pi(1-\cos 2 x) \cdot d x \\
& 2 \mathrm{I}=\frac{\pi}{2}\left[x-\sin 2 x \cdot \frac{1}{2}\right]_0^\pi \\
& I=\frac{\pi}{4}\left[\left(\pi-\frac{1}{2} \sin 2 \pi\right)-\left(0-\frac{1}{2} \sin 0\right)\right] \\
& =\frac{\pi}{4}[\pi] \quad \because \sin 0=0 ; \sin 2 \pi=0 \\
& =\frac{\pi^2}{4} \\
& \therefore \int_0^\pi x^2 \cdot \sin ^2 x \cdot d x=\frac{\pi^2}{4} \\
\end{aligned}
$

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.(3 Marks)" from Definite Integration→Definite Integration — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Represent the followinf families of curves by forming the corresponding differential equation:
$\text{y}=\text{e}^{\text{ax}}$

For the binary operation $\times _7 $ on the set $S = \{1, 2, 3, 4, 5, 6\},$ compute $3^{−1} \times _7 4.$

A wire of length $36$ metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.

If $\text{y}=3\text{e}^{2\text{x}}+2\text{e}^{3\text{x}}$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}-5\frac{\text{dy}}{\text{dx}}+6\text{y}=0$

Diffrentiate the following w.r.t.x

$\log \left[\frac{e^{x^2}(5-4 x)^{\frac{3}{2}}}{\sqrt[3]{7-6 x}}\right]$

Evaluate the following integrals:$\int\frac{\sin\text{x}-\cos\text{x}}{\sqrt{\sin2\text{x}}}\text{ dx}$

Show that $\text{AB}\neq\text{BA}$ in the following cases:
$\text{A}=\begin{bmatrix}-1&1&0\\0&-1&1\\2&3&4\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&2&3\\0&1&0\\1&1&0\end{bmatrix}$

In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}\text{kx}+5,&\text{if }\text{ x}\leq2\\\text{x}-1,&\text{if }\text{ x}>2\end{cases}$

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\text{a}+\text{x}}{1-\text{ax}}\Big)$

Evaluate $\triangle=\begin{vmatrix}0&\sin\alpha&-\cos\alpha\\-\sin\alpha&0&\sin\beta\\\cos\alpha&-\sin\beta&0 \end{vmatrix}$