Задача 7 (Невизначений інтеграл)

Знайти невизначений інтеграл  \int sin^{3}6xdx .

♦  \int sin^{3}6xdx=\int sin6x\cdot sin^{2}6xdx=

 =\int sin6x(1-cos^{2}6x)dx=\int (sin6x-sin6x\cdot cos^{2}6x)dx=

 =\int sin6xdx-\int sin6x\cdot cos^{2}6xdx=\begin{vmatrix} 6x=t\\ 6dx=dt\\ dx=\frac{dt}{6} \end{vmatrix}=

 =\frac{1}{6}\int sintdt-\frac{1}{6}\int sint\cdot cos^{2}tdt=

 =-\frac{1}{6}cost+C_{1}-\frac{1}{6}\int sint\cdot cos^{2}tdt=\begin{vmatrix} cost=z\\ -sintdt=dz\\ sintdt=-dz \end{vmatrix}=

 =-\frac{cos6x}{6}+C_{1}+\frac{1}{6}\int z^{2}dz=-\frac{cos6x}{6}+C_{1}+\frac{1}{6}\cdot \frac{z^{3}}{3}+C_{2}=

 =-\frac{cos6x}{6}+\frac{cos^{3}t}{18}+C=-\frac{cos6x}{6}+\frac{cos^{3}6x}{18}+C  .♦