Integrating Trig Functions

Some Results Have to be Memorised

There is no easy way to derive these results, so the best strategy is to commit them to memory.

\int \sin(x)dx=-\cos(x)+c

\int \cos(x)dx=\sin(x)+c

\int \sec^{2}(x)dx=\tan(x)+c

\int \cosec(x)\cot(x)dx=-\cosec(x)+c

\int \sec(x)\tan(x)dx=\sec(x)+c

\int \cosec^{2}(x)dx=-\cot(x)+c

If there is a coefficient before x, then you divide by the coefficient when you integrate (just like when integrating exponentials).

\int \sin(ax+b)dx=-\dfrac{1}{a}\cos(ax+b)+c

The other results follow in a similar way.

Other Results Can Be Derived

Other results can be derived using methods we have already learned.

Example: Find the integral of \tan(x).

Double Angle Formulas and Identities

We cannot integrate functions such as \sin^{2}x directly, but we can integrate functions like \sin(2x). This means that we can rearrange the double angle formulas to be able to integrate many more trigonometric functions. These are the most important ones to remember.

\sin^{2}(x)=\dfrac{1}{2}(1-\cos(2x))

\cos^{2}(x)=\dfrac{1}{2}(1+\cos(2x))

\sin(x)\cos(x)=\dfrac{1}{2}\sin(2x)

\dfrac{2\tan(x)}{1-\tan^{2}(x)}=\tan(2x)

We can use identities for the same purpose. For example, we cannot integrate \tan^{2}(x) directly, but we can integrate \sec^{2}(x), so the identities below can help.

\tan^{2}(x)=\sec^{2}(x)-1

\cot^{2}(x)=\cosec^{2}(x)-1