# Math Help - integrate sin^2xcos^4x

1. ## integrate sin^2xcos^4x

integrate:
$sin^2xcos^4x$

at the moment, I can only think of the cos^2x = 1/2cos2x + 1/2 thing, but I think that it will make the integration look really ugly

any other ideas for how to integrate this nicely?

2. Originally Posted by differentiate
integrate:
$sin^2xcos^4x$

at the moment, I can only think of the cos^2x = 1/2cos2x + 1/2 thing, but I think that it will make the integration look really ugly

any other ideas for how to integrate this nicely?

i would go with
substitution: t = cos^3(x)

then you need to integrate $\frac{1}{3}\cdot t^{\frac{4}{3}}$

3. Originally Posted by josipive
i would go with
substitution: t = cos^3(x)

then you need to integrate $\frac{1}{3}\cdot t^{\frac{4}{3}}$
Lovely idea - if only dt/dx would = 3 sin^2 x (!)

integrate sin&#x5e;2 x cos&#x5e;4 x - Wolfram|Alpha

4. Originally Posted by differentiate
integrate:
$sin^2xcos^4x$

at the moment, I can only think of the cos^2x = 1/2cos2x + 1/2 thing, but I think that it will make the integration look really ugly

any other ideas for how to integrate this nicely?

I would use the identity $\sin^2 \theta + \cos^2 \theta = 1$
$(1-\cos^2 x)cos^4 x = \cos^4(x) - \cos^6(x)$