13bii said or otherwise.
u can use 2 incomplete beta functions
\textstyle\int_0^\frac{\pi}{4}(\cos^4x+\sin^4x)\ \!dx=\frac{1}{2}(B_\frac{1}{2}(\frac{5}{2},\frac{1}{2})+B_\frac{1}{2}(\frac{1}{2},\frac{5}{2}))=\frac{1}{2}B(\frac{5}{2},\frac{1}{2})=\frac{3\pi}{16}