[sabre2th1]

I know that:

∫(Sin²tCos t )

= Sin³t/3 (using the f'(x)f(x) method)

However if I rewrote the integral as:


∫((1-Cos²t)Cos t )
=∫(Cos t - Cos³t )

How would I integrate the Cos³t, is this possible using the methods in C1-C4?

[Emissionspectra]

(Original post by sabre2th1)
I know that:

∫(Sin²tCos t )

= Sin³t/3 (using the f'(x)f(x) method)

However if I rewrote the integral as:


∫((1-Cos²t)Cos t )
=∫(Cos t - Cos³t )

How would I integrate the Cos³t, is this possible using the methods in C1-C4?

split it up using double angel formula.

[sabre2th1]

(Original post by Emissionspectra)
split it up using double angel formula.

How? I mean the double angle formulas deal with Cos²x

[Fing4]

(Original post by sabre2th1)
I know that:

∫(Sin²tCos t )

= Sin³t/3 (using the f'(x)f(x) method)

However if I rewrote the integral as:


∫((1-Cos²t)Cos t )
=∫(Cos t - Cos³t )

How would I integrate the Cos³t, is this possible using the methods in C1-C4?

You need to write in terms of and

hint

Spoiler:

Show

cox(3t) = cos(2t+t)

[sabre2th1]

(Original post by Fing4)
You need to write in terms of and

hint

Spoiler:

Show

cox(3t) = cos(2t+t)

Cos³x = Cos 3x ?

I didn't know that, could you show me how?

Thanks a lot

[Fing4]

(Original post by sabre2th1)
Cos³x = Cos 3x ?

I didn't know that, could you show me how?

Thanks a lot

It doesn't. You need to write an expression for cos^3(x) in terms of cos(3x) and cosx. I'll help you start.

Cox(3x)=cos(2x+x)=cos2xcosx-sin2xsinx


Edit: Why the neg? I know this isn't the easiest way to integrate cos^3(x) but this is what the OP asked.

[ThreeDaysGrace]

(Original post by sabre2th1)
I know that:

∫(Sin²tCos t )

= Sin³t/3 (using the f'(x)f(x) method)

However if I rewrote the integral as:


∫((1-Cos²t)Cos t )
=∫(Cos t - Cos³t )

How would I integrate the Cos³t, is this possible using the methods in C1-C4?

If you're ever asked to integrate Cos³t you should just use the fact that Cos³t = Cos t(1-Sin²t)

So Cos³t = Cos t-(Cost)Sin²t which can easily be integrated.


As a general rule if you have sin or cos to the power of an even number use the double angle formula.

If they're to the power of an odd number use the method above i.e. take out a factor of sin or cos and then split the rest into Sin²t or Cos²t and use Sin²t = 1-Cos²t)and Cos²t = 1-Sin²t