Question

Evaluate ${\int }_0^a{x}^3(ax-x^2{)}^{3/2}dx$

• A. $\frac{-9\pi a^7}{2048}$
• B. $\frac{3\pi a^7}{2048}$
• C. $\frac{9\pi a^7}{2048}$
• D. $\frac{9\pi a^7}{2345}$

Answer 1

Answer: (C)

$\frac{9\pi a^7}{2048}$

Given , ${\int }_0^a{x}^3(ax-x^2{)}^{3/2}dx$
Substitute $x=a{\sin }^{2}t$. $\Rightarrow$ $dx=2a\sin t\cos tdt$
$\Rightarrow$ ${\int }_0^{\frac{\pi }{2}}\left(a{\sin }^{2}t{)}^3\right(a\left(a{\sin }^{2}t\right)-\left(a{\sin }^{2}t{)}^2{)}^{3/2}\right(2a\sin t\cos tdt)$
$\Rightarrow$ ${\int }_0^{\frac{\pi }{2}}\left(a^6{\sin }^{9}t{\cos }^{3}t\right)\left(2a\sin t\cos tdt\right)$ , since $(1-{\sin }^{2}t)={\cos }^{2}t$
$\Rightarrow$ ${\int }_0^{\frac{\pi }{2}}\left(2a^7{\sin }^{10}t{\cos }^{4}t\right)dt$
Since ,${\int }_0^{\frac{\pi }{2}}\left({\sin }^{m}t{\cos }^{n}t\right)dt=\frac{(m-1)(m-3)......\times (n-1)(n-3)(n-5).......}{(m+n-1)(m+n-2)(m+n-4)..........}\times \frac{\pi }{2}$ if $m$ and $n$ are both even.
And here $m=10$ and $n=4$ So, we have
${\int }_0^{\frac{\pi }{2}}\left(2a^7{\sin }^{10}t{\cos }^{4}t\right)dt=\frac{9a^7\pi }{2048}$ as our answer.