Monday, October 13, 2014

Advanced Math Solutions – Integral Calculator, advanced trigonometric functions

In the previous post we covered substitution, but substitution is not always straightforward, for instance integrals involving powers of trig functions. We need first to transform the function into a more suitable form for substitution. We can do that using some manipulation and basic trig identities (we’ll show you all the tricks).

Here we will cover integrals of the form \sin^n(x)\cos^m(x).

The general rule to evaluate \int \sin^n(x)\cos^m(x)dx is as follows:
  • If n is odd, take sin(x) out, use the identity \sin^2(x)=1-\cos^2(x), and substitute u=\cos(x)
  • If m is odd, take cos(x) out, use the identity \cos^2(x)=1-\sin^2(x), and substitute u=\sin(x)
  • If both n and m are odd use either n or m
  • If both n and m are even use one of the half angle identities:
    • \sin^2(x)=\frac{1-\cos(2x)}{2}
    • \cos^2(x)=\frac{1+\cos(2x)}{2}
    • \sin(x)\cos(x)=\frac{\sin(2x)}{2}

Let’s see how it works, starting with an example where m is odd (click here):

From here simply apply the sum rule and substitute back.

Here’s another example where n is odd (click here):

One more example where both n and m are even (click here):

In the next post we will continue with integrals involving powers of tan and sec.