Antiderivatives of Basic Trigonometric Functions

We already know the derivatives of the six basic trig functions.

$\displaystyle\frac{d}{dx}\bigl(\sin(x)\bigr)=\cos(x)$

$\displaystyle\frac{d}{dx}\bigl(\cos(x)\bigr)=-\sin(x)$

$\displaystyle\frac{d}{dx}\bigl(\tan(x)\bigr)=\sec^2(x)$
 $\displaystyle\frac{d}{dx}\bigl(\cot(x)\bigr)=-\csc^2(x)$

$\displaystyle\frac{d}{dx}\bigl(\sec(x)\bigr)=\sec(x)\tan(x)$

$\displaystyle\frac{d}{dx}\bigl(\csc(x)\bigr)=-\cot(x)\csc(x)$

and the antiderivatives of two of them.
$\displaystyle\int\sin x\,dx=-\cos x+C$

$\displaystyle\int\cos x\,dx=\sin x+C$
 
In the video, we work out the antiderivatives of the four remaining trig functions.  Depending upon your instructor, you may be expected to memorize these antiderivatives.  The antiderivatives of tangent and cotangent are easy to compute, but not so much secant and cosecant.

$\begin{eqnarray}
\int\tan(x)\,dx&=&-\ln\bigl\lvert\cos(x)\bigr\rvert+C = \quad\ln\bigl\lvert\sec(x)\bigr\rvert+C \\
\displaystyle\int\cot(x)\,dx&=&\quad\ln\bigl\lvert\sin(x)\bigr\rvert+C \,= \,\,-\ln\bigl\lvert\csc(x)\bigr\rvert+C \\
\displaystyle\int\sec(x)\,dx&=& \quad\ln\bigr\lvert\sec(x)+\tan(x)\bigr\rvert+C \\
\displaystyle\int\csc(x)\,dx&=& -\ln\bigr\lvert\csc(x)+\cot(x)\bigr\rvert+C
\end{eqnarray}$