How To Integrate sin^5(x)
Published On :2020-12-06 20:41:00
begin{array}{l} smallint {sin ^5}x{rm{d}}x\\ smallint {sin ^4}{rm{x}}sin {rm{xdx}}\ smallint {left( {1 - {{cos }^2}x} right)^2}sin x{rm{d}}x\ By{rm{ }}Substitution;\ ;;;;;;;;;;;;;;;;;;Suppose; Rightarrow z = cosx;;;;;;;;;\ ;;;;;;;;;;;;;;;;;;;;;;;;;;;dz = - {rm{ }}sin{rm{ }}x{rm{ }}dx\ - smallint {left( {1 - {z^2}} right)^2}{rm{d}}z\ - smallint 1 - 2{z^2} + {z^4}{rm{d}}z;\ Rightarrow - left( {z - frac{{2{z^3}}}{3} + frac{{{z^5}}}{5}} right)\ Rightarrow - left( {cos x - frac{2}{3}{{cos }^3}x + frac{{{{cos }^5}x}}{5}} right) + C end{array}