What identity should I substitute for the integral of sin^4(x)*cos^4(x)?

[jaggtagg7]

can't figure this problem out for the life of me.
the intergral of:

sin^4(x)*cos^4(x)

any help as to what idenity i should sub in?

 

[Hurkyl]

If you still have your calculus textbook, there should be an entire section devoted specifically to integrals of this type. It's a good section to have handy as a reference, so I strongly suggest looking it up.


As for this particular example, have you noticed that the integrand is the same as (sin x cos x)^4 dx? (I'm assuming you meant to have a dx in there, though you didn't state it)

 

[jaggtagg7]

yes i have my book, but it doesn't help me figure out this problem which is what i need to do. what we've been doing is intergrals with one trig function to an odd power, and the other to an even. so the method has been substitue something for a trig power squared. i am lost as to what to do with two evens. i have tried using the same type of subsitutions but it has given me nothing. and i don't see where your suggestion leads me.

 

[Hurkyl]

The section really should have something on the case when they're both even powers as well... you should double check. I remember usually having to look through the section a couple times to find it in the text they used at the university to which I went.


As for my hint... if you stare at it, there should be something that leaps out and screams "do this" -- it's something that really should have come up before.

If not, once you figure it out or are told what it is, remember for the future.

 

[dextercioby]

$$\sin 2x =2\sin x \cos x$$ (1)

$$\cos^{2}x=\frac{1+\cos 2x}{2}$$ (2)

$$\sin^{2}x=\frac{1-\cos 2x}{2}$$ (3)

is all u need to solve this

$$\int \sin^{4}x\cos^{4}x \ dx$$

Daniel.