How to find $\int_0^1 x^4(1-x)^5dx$ quickly? This question came in the Rajshahi University admission exam 2018-19
Q) $\int_0^1 x^4(1-x)^5dx$=?
(a) $\frac{1}{1260}$
(b) $\frac{1}{280}$
(c)$\frac{1}{315}$
(d) None
This is a big integral (click on show steps):
$$\left[-\dfrac{\left(x-1\right)^6\left(126x^4+56x^3+21x^2+6x+1\right)}{1260}\right]_0^1=\frac{1}{1260}$$
It takes a lot of time to compute. How can I compute this quickly (30 seconds) using a shortcut?

We have
$$\begin{align}
\int_{0}^{1}x^4(1-x)^5\,dx
&=\int_{-1/2}^{1/2}(1/2+x)^4(1/2-x)^5\,dx\\
&=\int_{-1/2}^{1/2}(1/4-x^2)^4(1/2-x)\,dx\\
&=\int_{-1/2}^{1/2}(1/4-x^2)^4(1/2)\,dx\\
&\qquad+\int_{-1/2}^{1/2}(1/4-x^2)^4(-x)\,dx\\
&=\int_{-1/2}^{1/2}(1/4-x^2)^4(1/2)\,dx\\
&=\int_{0}^{1/2}(1/4-x^2)^4\,dx,
\end{align}$$
which is easily computed to $1/1260$.
