Recursive formulas and integration

Question

Using integration by parts find a recursive formula of $\int cos^n(x) dx$ and use it to find $\int cos^5 x dx$

I have no idea how to do this and my knowledge does include integration by parts etc. I just do not know what to do. Thanks!

Answer 1

Since they give you the trick (integration by parts), let us try $$u=\cos^{n-1}(x)\quad dv=\cos(x) \,dx$$ So, $$du=-(n-1)\cos^{n-2}(x)\,\sin(x)\,dx \quad v=\sin(x)$$ So, $$I_n=\int \cos^n(x)\,dx=\cos^{n-1}(x)\sin(x)+(n-1)\int \cos^{n-2}(x)\,\sin^2(x)\,dx$$ But $$\int \cos^{n-2}(x)\,\sin^2(x)\,dx=\int \cos^{n-2}(x)\,(1-\cos^2(x))\,dx=\int \cos^{n-2}(x)\,dx-\int \cos^{n}(x)\,dx$$ So, finally $$I_n=\cos^{n-1}(x)\sin(x)+(n-1)(I_{n-2}-I_n)$$

Isolate $I_n$ and you are done.