Integration by parts (2-x)cos(nPi/2)x?

Splint
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Homework Statement


Hi,

I'm doing fouier transforms and I'm not sure how to integrate (2-x)cos(nPi/2)x, (1,2). Anyone able to help me out? Even the indefinite integral would be fine.

Homework Equations



The Attempt at a Solution


I guess u would be (2-x) and dv would be cos(nPi/2)x dx. I'm not sure how to handle (nPi/2) since it is not a constant. Does (nPi/2) stay with x at all times?

Homework Statement


Homework Equations


The Attempt at a Solution

 
Last edited:
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n pi/2 IS a constant during the integration with respect to x.
Yes, it will be fine if you integrate by parts. How is the function exactly defined? Are you sure that you wrote the cosine function correctly? .

ehild
 
Last edited:
Thanks Ehild,

It's actually a half range expansion. The function is:

f(x)= (1, 0<x<1
(2-x, 1=<x=<2

So I believe the cosine function is correct. I won't try and integrate it right now but I should be ok with it.

Many thanks
Splint
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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