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 $tan(x)$.

$begin{aligned}int tan(x)dx&=intdfrac{sin(x)}{cos(x)}dx[1.2em]&=-intdfrac{-sin(x)}{cos(x)}dx[1.2em]&=-intdfrac{dfrac{d}{dx}(cos(x))}{cos(x)}dx[1.2em]&=-ln(|cos(x)|)+cend{aligned}$

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.

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

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

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

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

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.

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

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