_0^ x ^3 x ,dx = - Vidyadip

MCQ

$\int_0^\pi {x{{\sin }^3}x\,dx} = $

A
$\frac{{4\pi }}{3}$
✓
$\frac{{2\pi }}{3}$
C
$0$
D
None of these

✓

Answer

Correct option: B.

$\frac{{2\pi }}{3}$

b
(b) Let $I = \int_0^\pi {x{{\sin }^3}x\,dx} $....$(i)$

Also $I = \int_0^\pi {(\pi - x){{\sin }^3}x\,\,dx} $.....$(ii)$

Adding $(i)$ and $(ii),$ we get

$2I = \pi \int_0^\pi {{{\sin }^3}x} \,\,dx = \frac{\pi }{4}\int_0^\pi {\{ 3\sin x - \sin 3x\} dx} $

$ = \frac{\pi }{4}\left[ { - 3\cos x + \frac{{\cos 3x}}{3}} \right]_0^\pi $

$= \frac{\pi }{4}\left[ {3 - \frac{1}{3} + 3 - \frac{1}{3}} \right] = \frac{{4\pi }}{3}$

Hence, $I = \frac{{2\pi }}{3}$.

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 "M.C.Q (1 Marks)" from 7.2 definite integral→7.2 definite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+2 \hat{j}+3 \hat{k}$. Then the vector product $(\vec{a}+\vec{b}) \times((\vec{a} \times((\vec{a}-\vec{b}) \times \vec{b})) \times \vec{b})$ is equal to:

The direction ratios of the diagonal of the cube joining the origin to the opposite corner are (when the 3 concurrent edges of the cube are coordinate axes).

If $f(x)=2|x-2|-3|x-3|$ then $f(x)$ equals to, for 2 $< x< 3$.

Let there be three independent events $E _{1}, E _{2}$ and $E _{3}$. The probability that only $E _{1}$ occurs is $\alpha$, only $E _{2}$ occurs is $\beta$ and only $E _{3}$ occurs is $\gamma .$ Let $'p'$ denote the probability of none of events occurs that satisfies the equations $(\alpha-2 \beta) p =\alpha \beta$ and $(\beta-3 \gamma) p =2 \beta \gamma .$ All the given probabilities are assumed to lie in the interval $(0,1)$

Then, $\frac{\text { Probability of occurrence of } E _{1}}{\text { Probability of occurrence of } E _{3}}$ is equal to ..........

The number of points in $(-\infty,\infty)$ for which $\text{x}^{2}-\text{x}\sin\text{x}-\cos\text{x}=0,$ is:

6
4
2
None of the above

$\sin ^{-1}\left(-\frac{1}{2}\right)+\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)=$___________.

If the area above the $x -$ axis, bounded by the curves $y = {2^{kx}}$ and $x = 0$ and $x = 2$ is $\frac{3}{{\ln 2}},$ then the value of $k$ is

Suppose $\quad f : R \rightarrow(0, \infty)$ be a differentiable function such that $5 f ( x + y )= f ( x ) \cdot f ( y ), \forall x , y \in R$. If $f(3)=320$, then $\sum \limits_{n=0}^5 f(n)$ is equal to :

Let $R = \{ (3,\,3),\;(6,\;6),\;(9,\,9),\;(12,\,12),\;(6,\,12),\;(3,\,9),(3,\,12),\,(3,\,6)\} $ be a relation on the set $A = \{ 3,\,6,\,9,\,12\} $. The relation is

The degree of differential equation $\frac{{{d^2}y}}{{d{x^2}}} + {\left( {\frac{{dy}}{{dx}}} \right)^3} + 6y = 0$ is