Integration of Trigonometric Functions

Example 1:

Solution:

• $$\int \sin^4 x \cos x\, dx = \int (\sin x )^4\cos x \, xdx$$
• $$\textup{Let}\,\, u=\sin x$$
• $$du = \cos x dx$$
• $$\int \sin^4 x \cos x\, dx = \int u^4 \,du $$
• $$= \frac{u^{4+1}}{4+1}+C$$
• $$= \frac{u^{5}}{5}+C$$
• $$= \frac{1}{5} \, u^5+C$$
• $$= \frac{1}{5} \, (\sin x)^5+C$$
• $$= \frac{1}{5} \, \sin^5 x+C$$