Is integration by parts the best method for $\int_0^1 x^3(1-x)^6 dx$?

Question

It came up when finding a constant such that the integral is equal to 1 and thus behaves like a pdf. I used the parts method but have made an error, just curious how others might approach the problem.

Answer 1

The beta function is the best idea.

$$\beta(a, b) = \int_{0}^{1} x^{a-1}(1-x)^{b-1} dx$$

For here $I$, let $a = 4, b = 7$

Using:

$$\beta(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$

$$\beta(4, 7) = \frac{\Gamma(4)\Gamma(7)}{\Gamma(11)} = \frac{3!6!}{10!}$$

$$= \frac{3\cdot2\cdot1}{10\cdot9\cdot8\cdot7} = \frac{1}{840}$$

Answer 2

This way might be faster, but it ultimately depends on your personal preference.

Let $u = 1-x$. Then the integral becomes:

$$\int (u-1)^3u^6 \ \text{d}u$$

And that expansion is much easier to work with than the original!

Answer 3

Along the lines of Kaj, I think the fastest is to just expand: $$ \int x^3(1-x)^6 = \int (1-u)^3u^6=\int u^6-3u^7+3u^8-u^9=\frac{1}{7}-\frac{3}{8}+\frac{3}{9}-\frac{1}{10}. $$

Answer 4

By symmetry,

\begin{align} \int_0^1 x^3(1-x)^6\,\mathrm{d}x &= -\int_1^0x^6(1-x)^3\mathrm{d}x\\ &=-\int_0^1 (x-1)^3x^6\,\mathrm{d}x \\ &=-\int_0^1 x^9-3x^8+3x^7-x^6 \,\mathrm{d}x\\ &=-\left(\frac{1}{10}-\frac{1}{3}+\frac{3}{8}-\frac{1}{7}\right)+0\\\\ &=-\frac{84-280+315-120}{840}\\ &=\boxed{\displaystyle\frac{1}{840}} \end{align}

Answer 5

If we realize the integral is part of the Beta Distribution: $$\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1},0<x<1$$ We find $\alpha=4,\beta=7$. We also know the integral of any pdf equals $1$ and thus this our coefficient must be the inverse of the result.

Answer 6

using that $$x^3(1-x)^6=x^9-6 x^8+15 x^7-20 x^6+15 x^5-6 x^4+x^3$$ we can calculate the integral without the beta function, we get $$\int_{0}^{1} x^9-6 x^8+15 x^7-20 x^6+15 x^5-6 x^4+x^3dx=\frac{1}{840}$$