## anonymous one year ago Use a substitution to find.....

1. anonymous

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2. anonymous

So i choose $u=\cos 4t$ Derrivative gives $\frac{ du }{ dt }=-4\sin 4t$ Then Rewrite $\sin 4t dt = \frac{ du }{ 4 }$ and finally i got to this $\frac{ 1 }{ 4 } \int\limits_{}^{} \sin ^{2} u dt$

3. anonymous

Have I done correctly so far or no? What/where do I need to do changes.

4. mathmate

You can do the substitution in two steps, p=4t, and followed by q=sin(p), or as you suggested, you can do it in one step: u=sin(4t). I suggest u=sin(4t) because eventually the integration is simpler than u=cos(4t). I will first work it out using your suggested substitution. u=cos(4t); du=-4sin(4t)dt then I=$$\int sin^2(4t)cos(4t)dt$$ =-$$\int (1/4)sin(4t)udu$$ =-(1/4)$$\int sqrt(1-u^2)udu$$ which makes the integral more awkward than necessary. If we proceed with u=sin(4t), life will be a little easier, as you can see below: u=sin(4t), du=4cos(4t)dt, or cos(4t)dt = (1/4)du I=$$\int sin^2(4t)cos(4t)dt$$ =(1/4)$$\int u^2du$$ =$$\Large \frac{u^3}{12}$$ =$$\Large\frac{sin^3(4t)}{12}$$ Rule of thumb: use substitution u=sin(x) when the integrand is $$sin^n(x)cos(x)dx$$ or use substitution u=cos(x) when the integrand is $$cos^n(x)sin(x)dx$$

5. mathmate

@Xlegalize

6. anonymous

thanks, I solved it. But yeah, I see were I did my wrongs. THanks for ur time.