<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] Personally I'm somewhat inclined to use the fifth row of Pascal's triangle: $$  
                 1\\
                1\quad 1 \\
               1\quad  2\quad 1\\
              1\quad  3\quad 3\quad 1\\
            1 \quad 4\quad  6\quad  4 \quad 1\\
            1\quad  5\quad  10\quad 10\quad 5 \quad 1$$
Thus $$(1-x)^5=1-5x^1+10x^2-10x^3+5x^4-x^5$$.
So, $$x^4-5x^5+10x^6-10x^7+5x^8-x^9$$ is the integrand.
We get $$ 1/5-5/6+10/7-10/8+5/9-1/10=-19/30+10/56+41/90=-16/90+10/56=-8/45+5/28=1/1260$$.
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