Should we integrate by parts?

References:

• Beta function
• Gamma function

---

From the definition of the beta function

$B(m,n) = \displaystyle 2 \int_0^{{\pi}/{2}} \sin^{2m-1} \theta \cos^{2n-1} \theta \ \operatorname{d\theta} = \dfrac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)}$

By comparison, we get our integral as

$\begin{aligned} \dfrac 12 \cdot B \left( \dfrac{1001}{2} , \dfrac{1}{2} \right) &= \dfrac 12 \cdot \dfrac{\Gamma \left(\frac{1001}{2}\right) \Gamma \left(\frac{1}{2}\right)}{\Gamma(501)} \\ &= \dfrac 12 \cdot \dfrac{\displaystyle \prod_{n=2}^{500} (2n-1) \cdot \frac{1}{2^{499}} \cdot \Gamma^2 \left( \frac 12 \right)}{(500)!} \\ &= \dfrac{\pi}{2} \cdot \dfrac{\displaystyle \prod_{m=1}^{1000} m}{2^{1000} \cdot (500)! \cdot (500!)} \\ &= \dfrac{\pi}{2} \cdot \dfrac{(1000)!}{2^{1000} {(500)!}^2} \end{aligned}$

$\implies a+b = 1000+500 = \boxed{1500}$

Similar solution to Tapas Muzumdar 's

$\begin{aligned} I & = \int_0^\frac \pi 2 \sin^{1000} \theta \ d\theta \\ & = \int_0^\frac \pi 2 \sin^{1000} \theta \cos^0 \theta \ d\theta \\ & = \frac 12 B\left(\frac {1001}2, \frac 12 \right) & \small \color{#3D99F6} B(m,n) \text{ is beta function.} \\ & = \frac 12 \cdot \frac {\Gamma \left(\frac {1001}2 \right) \Gamma \left(\frac 12 \right)}{\Gamma \left(501\right)} & \small \color{#3D99F6} \Gamma(s) \text{ is gamma function.} \\ & = \frac 12 \cdot \frac {\frac {999!!}{2^{500}} \sqrt \pi \cdot \sqrt \pi}{500!} \\ & = \frac \pi 2 \cdot \frac {1000!}{2^{1000}(500!)^2} \end{aligned}$

$\implies a + b = 1000 + 500 = \boxed{1500}$

---

Reference

• Beta function
• Gamma function

The given integral is a form of the Wallis' integrals discovered by the mathematician John Wallis. We can easily obtain a recurrence relation on the general integral for even numbers and obtain the values of a and b