Integration of Trigonometric Functions

Example 1:

$$\int \tan x \, dx $$

Solution:

\begin{align*} \int \tan x \, dx &= \int \frac{\sin x }{\cos x }\, dx\\\\ \textup{Let}\,\, u&=\cos x\\ du &= -\sin x dx\\ -du &= \sin x dx\\\\ \int \tan x\, dx &= \int \frac{-du}{u}\\ &=- \int \frac{1}{u}\,du\\ &= -\ln \lvert u \lvert+C\\ &= -\ln \lvert \cos x \lvert+C\\ &= \ln 1 -\ln \lvert \cos x \lvert+C\\ &= \ln \frac{1}{\lvert \cos x \lvert}+C\\ &= \ln \lvert \sec x \lvert+C\\ \end{align*}

Example 2:

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

Solution:

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