Clever integral and formula

Question

Find $$\int \sin(2020x)\sin^{2018} (x) dx$$ and use it to derive a general formula.

Answer 1

https://www.wolframalpha.com/input/?i=integral+sin%282020+x%29+sin%5E%282018%29%28x%29+dx

To elaborate: let $$f(x)=\frac{\sin^{2k+1}(x) \sin((2k+1)x)}{2k+1}$$ Then, $$\begin{aligned} f'(x)&=\frac{(2k+1)\sin^{2k}(x)\cos(x) \sin((2k+1)x) + (2k+1)\sin^{2k+1}(x) \cos((2k+1)x)}{2k+1}\\ &=\sin^{2k}(x)\Big(\cos(x) \sin((2k+1)x) + \sin(x) \cos((2k+1)x)\Big)\\ &=\sin^{2k}(x)\Big(\sin((2k+2)x)\Big) \end{aligned}$$ Where the third equality follows from: https://www.cut-the-knot.org/Curriculum/Algebra/SineInRhombus.shtml