As in the title: how does one integrate $$ \int\sin^2(x)\cos^4(x)dx? $$
Any hints? Integration by parts doesn't seem to be a reasonable technique here.
[Math] Integrate $\int\sin^2(x)\cos^4(x)dx$
fourier seriesintegration
fourier seriesintegration
As in the title: how does one integrate $$ \int\sin^2(x)\cos^4(x)dx? $$
Any hints? Integration by parts doesn't seem to be a reasonable technique here.
Best Answer
In idea for you to develop:
$$\sin x\cos x=\frac12\sin 2x\implies \sin^2x\cos^4x=\left(\frac12\sin2x\right)^2\cos^2x=$$
$$=\frac14\sin^22x\left(\frac12\left(\cos2x+1\right)\right)=\frac18\cos2x\sin^22x+\frac18\sin^22x$$
Observe now that
$$\left(\sin2x\right)'=2\cos 2x,\,\text{ and}\;\;\int\sin^22x\,dx=\frac12\int\sin^2u\,du=\frac14\left(u-\sin u\cos u\right)=...$$