Question

Evaluate the integral: $\int_{0}^{π/2} \sqrt{\sin \phi}\cos^5 \phi\, d\,\phi$

Answer 1

Let $I$=$\int_{0}^{π/2} \sqrt{\sin \phi}\cos^5 \phi\, d\,\phi$=$\int_{0}^{π/2} \sqrt{\sin \phi}\cos^4 \phi\, d\,\phi$

Also, let $\sin \phi=t\Rightarrow \cos \phi d\phi=dt$

When $\phi$ =0,t=0 and when $\phi=\frac{\pi}{2},t=1$

∴ $I=\int^1_0\sqrt{t}(1-t^2)dt$

= $I=\int^1_0t^{\frac{1}{2}}(1+t^4-2t^2)dt$

=$I=\int^1_0\bigg[t^{\frac{1}{2}}+t^{\frac{9}{2}}-2t^{\frac{5}{2}}\bigg]dt$

=$\bigg[\frac{t^{\frac{3}{2}}}{\frac{3}{2}}+\frac{t^{\frac{11}{2}}}{\frac{11}{2}}-\frac{2t^{\frac{7}{2}}}{\frac{7}{2}}\bigg]^1_0$

=$\frac{2}{3}+\frac{2}{11}-\frac{4}{7}$

=$\frac{154+42-132}{231}$

=$\frac{64}{231}$