Oneven machten van sinus of cosinus
Oneven machten van sinus of cosinus zijn makkelijk te integreren.Voorbeeld 1
\int {\sin ^3 x\,\,dx = } \int {\sin ^2 x\sin x\,\,dx} = \int {\left( {1 - \cos ^2 x} \right)d( - \cos x)}
Kies
u = -\cos (x)
\int {1 - u^2 \,du = u - \frac{1}{3}u^3 + C = - \cos x + \frac{1}{3}\cos ^3 x + C}
Voorbeeld 2
\begin{array}{l} \int{\cos ^5 x} \,dx =\int{\cos ^4 x \cos x} \,dx =\int{\left( {1 - \sin ^2 x} \right)^2 } d\left( {\sin x} \right) = \\ \int {\left( {1 - u^2 } \right)^2 du} = \int {\left( {1 - 2u^2 + u^4 } \right)du} = \\ u - \frac{2}{3}u^3 + \frac{1}{5}u^5 = \sin x - \frac{2}{3}\sin ^3 x + \frac{1}{5}\sin ^5 x \\ \end{array}