Integration of Trigonometric Functions

Example 1:

$$\int \sin^7 x \cos x\, dx $$

Solution:

\begin{align*} \int \sin^7 x \cos x\, dx &= \int (\sin x )^7\cos x \, xdx \\\\ \textup{Let}\,\, u & =\sin x \\ du & = \cos x \, dx \\\\ \int \sin^7 x \cos x\, dx & = \int u^7 \,du \\ & = \frac{u^{7+1}}{7+1}+C \\ & = \frac{u^{8}}{8}+C \\ & = \frac{1}{8} \, u^8+C \\ & = \frac{1}{8} \, (\sin x)^8+C \\ & = \frac{1}{8} \, \sin^8 x+C \\ \end{align*}

Example 2:

$$\int \cos^4 x \sin x\, dx $$

Solution:

\begin{align*} \int \cos^4 x \sin x\, dx &= \int (\cos x )^4 \sin x \, xdx \\\\ \textup{Let}\,\, u & =\cos x \\ du & =-\sin x \, dx \\ -du & = \sin x \, dx \\\\ \int \cos^4 x \sin x\, dx & = \int u^4 \, (-du) \\ & = -\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} \, (\cos x)^5+C \\ & = -\frac{1}{5} \, \cos^5 x+C \\ \end{align*}

Example 3:

$$\int \sin 5x \, dx $$

Solution:

\begin{align*} \int &\sin 5x \, dx \\\\ \textup{Let}\,\, u & =5x \\ du & = 5dx \\ \frac{du}{5} & = dx \\\\ \int \sin 5x \, dx & = \int \sin u \,\frac{du}{5} \\ & = \frac{1}{5} \int \sin u \,du \\ & = \frac{1}{5} (-\cos u) +C \\ & = -\frac{1}{5} \cos u +C \\ & = -\frac{1}{5} \cos 5x +C \\ \end{align*}