<s>[INST] 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?
 [/INST] Maybe it is not the fastest way to solve it, but it is faster than computing the original integral:
\begin{align*}
I = \int_{0}^{1}x^{4}(1 - x)^{5}\mathrm{d}x = \int_{0}^{1}(1 - x)^{4}x^{5}\mathrm{d}x & \Rightarrow 2I = \int_{0}^{1}x^{4}(1-x)^{4}(x + (1 - x))\mathrm{d}x\\\\
& \Rightarrow 2I = \int_{0}^{1}x^{4}(1-x)^{4}\mathrm{d}x\\\\
& \Rightarrow 2I = \int_{0}^{1}x^{4}(1 - 4x + 6x^{2} - 4x^{3} + x^{4})\mathrm{d}x\\\\
& \Rightarrow 2I = \int_{0}^{1}(x^{4} - 4x^{5} + 6x^{6} - 4x^{7} + x^{8})\mathrm{d}x\\\\
& \Rightarrow 2I = \frac{1}{5} - \frac{2}{3} + \frac{6}{7} - \frac{1}{2} + \frac{1}{9}\\\\
& \Rightarrow I = \frac{1}{1260}
\end{align*}
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