The Beta function is defined by

$$B(p, q):=\int_{0}^{1} t^{p-1}(1-t)^{q-1} d t=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}$$

where $\operatorname{Re} p>0, \operatorname{Re} q>0$, and $\Gamma$ is the Gamma function.

(a) By using a suitable substitution and properties of Beta and Gamma functions, show that

$$\int_{0}^{1} \frac{d x}{\sqrt{1-x^{4}}}=\frac{[\Gamma(1 / 4)]^{2}}{\sqrt{32 \pi}}$$

(b) Deduce that

$$K(1 / \sqrt{2})=\frac{4[\Gamma(5 / 4)]^{2}}{\sqrt{\pi}}$$

where $K(k)$ is the complete elliptic integral, defined as

$$K(k):=\int_{0}^{1} \frac{d t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2} t^{2}\right)}}$$

[Hint: You might find the change of variable $x=t\left(2-t^{2}\right)^{-1 / 2}$ helpful in part (b).]