D-0-pi-2-sin-4-x-cos-5-x-dx-

Question Number 134708 by bramlexs22 last updated on 06/Mar/21

D = \int_{0}^{π / 2} \sin^{4} x \cos^{5} x dx

Answered by john_santu last updated on 06/Mar/21

D = \int_{0}^{π / 2} \sin^{4} x {(1 - \sin^{2} x)}^{2} \cos x d x

l e t u = \sin x, \begin{array}{cc} u = 1 (u p p e r l i m i t) \\ u = 0 (l o w e r l i m i t) \end{array}

D = \int_{0}^{1} u^{4} (u^{4} - 2 u^{2} + 1) d u

D = \int_{0}^{1} (u^{8} - 2 u^{6} + u^{4}) d u

D = {[\frac{u^{9}}{9} - \frac{2 u^{7}}{7} + \frac{u^{5}}{5}]}_{0}^{1}

D = \frac{1}{9} - \frac{2}{7} + \frac{1}{5} = \frac{35 - 90 + 63}{315}

D = \frac{8}{315} ∙

Answered by Dwaipayan Shikari last updated on 06/Mar/21

\int_{0}^{\frac{π}{2}} {s i n}^{4} x {c o s}^{5} x d x, G e n e r a l l y \int_{0}^{\frac{π}{2}} {s i n}^{m} x {c o s}^{n} x d x = \frac{Γ (\frac{m + 1}{2}) Γ (\frac{n + 1}{2})}{2 Γ (\frac{m + n}{2} + 1)}

= \frac{Γ (\frac{5}{2}) Γ (3)}{2 Γ (\frac{11}{2})} = \frac{\frac{3}{4} \sqrt{π} . 2}{2 . \frac{9}{2} . \frac{7}{2} . \frac{5}{2} . \frac{3}{4} \sqrt{π}} = \frac{8}{315}