## anonymous 3 years ago integrate xcos^2(8x)dx

1. anonymous

$\int\limits_{}^{}xcos^2(8x)dx$

2. anonymous

I think I might need a half angle formula? I'm still confused though...

3. anonymous

Yep, power-reducing formula, and then integration by parts. Use the fact that: $\cos^2(x)=\frac{1+\cos(2x)}{2}$

4. anonymous

okay, so I've got it down to $\int\limits_{}^{}x \frac{ 1 }{ 2 }(1+\cos(16x))dx$

5. anonymous

is that right?

6. anonymous

To make life easier, think of it in terms of powers of e: $e^{i\theta}=\cos(\theta)+i\sin(\theta)$If you wish.

7. anonymous

Yes, that is.

8. anonymous

can you give me a hint what to do next please?

9. anonymous

Sure: try to integrate it using integration by parts. Do you know how to do this...?

10. anonymous

Yes I do. Should I use u=1+cos(16x) and dv=x ?

11. anonymous

dv=x dx

12. anonymous

Try the other way around. Since we wish to reduce it to an integrable series. E.g. we can integrate $$\sin(l)$$, but not $$\frac{x^2}{2}\sin(l)$$, without integration by parts.

13. anonymous

Try doing the following case: $dv=(1+\cos(16))dx,\;v=\cdots\\ u=x,\;du=dx$

14. anonymous

oh, gotcha. okay just a sec

15. anonymous

Quick tip, remember, intuitively, what integration by parts does: For some functions $$f, g$$, integration by parts turns the problem: $\int f'\cdot g\;dx$Into the problem: $\int f\cdot g'\;dx$Think about this a little bit and it will become much easier to understand its use.

16. anonymous

$x(x+\frac{ \sin(16x) }{ 16 })-\int\limits_{}^{}x+\frac{ \sin(16x) }{ 16 } dx$

17. anonymous

is that right so far?

18. anonymous

Yessir/ma'am. Although don't forget the constant $$\frac{1}{2}$$ that was at the beginning of the problem.

19. anonymous

yep I left that hanging around somewhere. Okay, so without the 1/2 from before I got $x^2+xsin(16x)/16 -\int\limits_{}^{}x+\frac{ \sin(16x) }{ 16 }dx$

20. anonymous

is that right?

21. anonymous

which would then be $x^2+\frac{ xsin(16x) }{ 16 }-\frac{ x^2 }{ 2 }-\frac{ \cos(16x) }{ 32 }$

22. anonymous

oops that last one should be plus, not minus

23. anonymous

ahhh nope that's not right just a sec

24. anonymous

You're really close, remember that the denominator has to multiply by 16, not add, and that the integral of sin is -cos, not +cos. But, you've got the jist. The devil is in the details.

25. anonymous

Okay, I got the right answer. Thanks so much for the help!! :)

26. anonymous

Sure thing, have a good one.