How can I integrate (Sin^4(x)*Cos^4(x)) without a calculator?

[Alabran]

Homework Statement

Find the integral of (Sin^4(x)*Cos^4(x)) in respect to x without using a calculator.

Homework Equations

Sin^2(x) + Cos^2(x) = 1
Sin^2(x) = (1-Sin(2X))/2
Cos^2(x) = (1+Sin(2X))/2
Sin(2x) = 2Sin(x)Cos(x)
Cos(2x) = Cos^2(x) - Sin^2(x)

The Attempt at a Solution

I've been attempting to manipulate the equation so-as-to isolate a chain rule factor so I can use the U subsitution method on the problem. Unfortunately, my efforts so far have been fruitless.

 

[Mindscrape]

Unfortunately these are both even powers, so you won't be able to get a u-substitution out of it (at least not a trivial one). You can do some sneaky algebra to get it into an easily integrable form, or just put it all in terms of sine or cosine to a power, and use a reduction formula.

 

[malawi_glenn]

Sin^4(x)*Cos^4(x) = (sin(x)cos(x))^4 = ((1/2)Sin(2x))^4

than i would use Eulers formulas:

sin(b) = (e^ib-e^-ib)/(2i)

cos(b) = (e^ib+e^-ib)/(2)

Have also tried using Eulers formulas?
My favorite in integrating trigonometric functions.

 

[malawi_glenn]

so you can express the Sin^4(x)*Cos^4(x) as

(-1/16)*(2cos8x - 8cos4x + 6)

can have done some errors, it is very late now in sweden hehe

 

[Alabran]

I haven't tried the Euler's forumula method, though I doubt that is what is intended.

In class, the answer to that and several other similar problems were put up in random order to show where we should be at the end. The ultimate answer was quite clean, just a sum or difference of trigonometric fractions of different powers. I don't quite remember the exact answer, though I believe the denominators were 128.

Thank you all for your help in the meantime.

 

[malawi_glenn]

{should be (-1/16^2)*(2cos8x - 8cos4x + 6)

yes:)

Eulers formula ALWAYS work, i have never seen a trigonometric function that it don't works on. You don't have to remember identities and so on, just Euelers two formulas hehe

 

[Alabran]

Thank you for your help Malawi, that's a start for me, though I don't believe that's the answer my teacher is looking for. I'll try manipulating the Euler's formula some more.

 

[malawi_glenn]

No it is not the answer, i just said that you can express (sin^4(x)cos^4(x)) as that..

and yeah, forget the (-1/16^2) , shoulb be (1/16^2) = 1/256