How Do I Integrate cos(x) and sin(x) with higher powers?

The answer to this question depends on one thing, whether this power is odd or even.

For even powers we use the identity cos(2x) = 2cos^2(x)-1 = 1-2sin^2(x). So for example, the integral of cos^2(x) is found by substituting this identity to give us the integral of 0.5(cos(2x)-1) which can be easily solved through basic integration principles to give us: 0.25sin(2x)-0.5x +c.

For odd powers the integration is a little harder to get used to, this is because it relies on lateral thinking rather than simply applying principles. For example to integrate sin^3(x) we make use of the sin^2(x)+cos^2(x)=1 identity. Once this identity has been used this gives us sin(x)(1-cos^2(x)). Once the bracket is multiplied out this integral can then be divided into two integrals for ease, giving us,int(sin(x)) minus int(sin(x)cos^2(x)). The first is simple to solve through basic principles of trigonometric calculus, the result is -cos(x). The second part is where lateral thinking becomes important. To solve this we must consider what we need to differentiate to create this result, thinking in reverse. Through practice students will eventually be able to pick up on this. As the power is ^2 we know the power of the equation that's been differentiated is ^3. Therefore we could guess the solution to the integral is cos^3(x). However, when we differentiate this we get 3sin(x)cos^2(x) which is 3 times larger than our original integral. Therefore the solution to the integral is 3 times smaller than we assumed. Therefore the solution to the overall integral is -cos(x)-1/3cos^3(x)+c.