How to integrate with reduction formula?

Question

$$\begin{array}{|c|c|} \hline \text{Integral} & \text{Reduction formula} \\ \hline\displaystyle I_{m,n}=\int\sin^max\cos^nax\,\mathrm dx & I_{m,n}=\begin{cases}-\tfrac{\sin^{m-1}ax\cos^{n+1}ax}{a(m+n)}+\tfrac{m-1}{m+n}I_{m-2,n}\\\tfrac{\sin^{m+1}ax\cos^{n-1}ax}{a(m+n)}+\tfrac{n-1}{m+n}I_{m,n-2}\end{cases} \\ \hline \end{array}$$

How to integrate $(\cos x)^n\cdot(\sin x)^m$ with reduction formula. The actual steps please in terms of $(n-2,m)$ and in terms of $(n, m-2)$ like in the table shown. I'm stuck halfway.

Thanks

Answer 1

Consider the example with $m = 3$ and $n= 2$.

We have,

$$I_{m,n} = I_{3,2} = \int \sin^3(ax)\cos^2(ax) \,d x= -\frac{\sin^{3-1}(ax)\cos^{2 + 1}(ax)}{a(3 + 2)} + \frac{3-1}{3 + 2}I_{1,2}.$$

Here I have applied the first reduction formula. I have not reduced the result far enough to be complete. However, I can now continue until the $I_{1,2}$ term disappears. Maybe apply the second reduction next?

Every example would proceed in this way. You must iteratively apply the formula until the integral dissapears or becomes one that you know how to handle.

As an aside for $m \ge 3$ you can also make use of the identity $\sin^2(x) + \cos^2(x) = 1$,

$$\int sin^m(ax) \,d x = \int \sin^{m-2}(ax)(1-\cos^2(ax)) \,d x .$$

In addition, the double-angle substitutions can be useful as well.

These are $$\cos^2(x) = \frac{1}{2}(1 + \cos(2x))$$ and $$ \sin^2(x) = \frac{1}{2}(1 - \cos(2x)).$$