What is wrong with my Integration?

**Henryt999** · January 14th 2010, 12:31 PM

The question is: Give an exakt value to P so that:
$Int<img src=$ P-sin^3v):dv = 1" alt="Int

P-sin^3v):dv = 1" /> Int: limits are from $0->pi$

$ sin3v = 3sinv-4sin^3v -> 4sin^3v = 3sinv-sin3v -> sin^3v = 3/4sinv-1/4sin3v $

$Int(P-3/4sinv+1/4sin3v):dv$ from $0;pi$
$ Int(P-3/4sinv+1/4sin3v):dv $ = $(Pv+3/4cosv-3/4cos3v)$

$P(pi)+3/4+3/4-3/4-3/4 = 1$ -> $P(pi) = 1$ but the answer is $P = 7/3pi$

**General** · January 14th 2010, 12:51 PM

Sorry I did not follow your solution.
Anyway, Here is the solution:
$\int_{0}^{\pi} ( P - sin^3(v) ) dv = 1$
$P\int_{0}^{\pi} dv - \int_{0}^{\pi} sin^3(v) dv = 1$
$(P)(\pi) - \int_{0}^{\pi} ( 1-cos^2(v) ) sin(v) dv = 1$
For the integral substitute
$ t=cos(v) $
$ dt=-sin(v) dv $
$ t(\pi)=-1 $
$ t(0)=1 $
$ (P)(\pi) + \int_{1}^{-1} ( 1 - t^2 ) dt = 1 $
$ (P)(\pi) - \int_{-1}^{1} ( 1 - t^2 ) dt = 1 $
since $f(t)=1-t^2$ is an even function, the last equation can be writte as follows:
$ (P)(\pi) - 2 \int_{0}^{1} ( 1 - t^2 ) dt = 1 $
Integrate and evaluate, to get:
$ (P)(\pi) - 2 [ 1 - \frac{1}{3} ] = 1 $
$ (P)(\pi) - 2 (\frac{2}{3}) = 1 $
$ (P)(\pi) - \frac{4}{3} = 1 $
$ (P)(\pi) = \frac{7}{3} $
$ P = (\frac{7}{3}) (\frac{1}{\pi}) $
$ P=\frac{7}{3\pi} $

----

Use \pi to write pi`s symbol in LaTeX.

**Henryt999** · January 14th 2010, 01:01 PM

I´ll give you Thanks for sure.

But Oh My God is that what I have to do to solve it?
Haven´t learned a thing about Integration by substitution yet, I checked the wikipedia for info on the subject, but I think it will take some time to melt. You wouldn´t happen to have a e-book/book to recommend on the subject? Or know anywhere I can find some info on how to integrate by substitution?
Thanks you for your effort

**General** · January 14th 2010, 01:21 PM

Originally Posted by Henryt999

I´ll give you Thanks for sure.

But Oh My God is that what I have to do to solve it?
Haven´t learned a thing about Integration by substitution yet, I checked the wikipedia for info on the subject, but I think it will take some time to melt. You wouldn´t happen to have a e-book/book to recommend on the subject? Or know anywhere I can find some info on how to integrate by substitution?
Thanks you for your effort

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