Help with an integration by parts problem

[stanners]

integration problem :(

Homework Statement

int from 1 to 1/4 of [cos(pi*sqrt(t))] / sqrt(t) dt

Homework Equations

The Attempt at a Solution

I tried using integration by parts

used u = cos pi (sqrt(t)) and dv = sqrt(t), but got a really messy number as int of vdu

so I tried u = sqrt t, du = 1/2sqrt(t)
dv = cos pi*sqrt(t) , v = sin pi sqrt(t) * pi/2sqrt(t)


So This doesn't seem right, how would I approach this problem?

 

[tronter]

You want $$\int_{1}^{\frac{1}{4}} \frac{\cos \pi \sqrt{t}}{\sqrt{t}} \; dt$$

 

[stanners]

Yeah, but integration by parts didn't work, and I don't see how I can use substitution, any help?

 

[Reshma]

The change of variable method is correct.
Put $u = \sqrt t$
$2du = \frac{dt}{\sqrt t}$
Substituting this will simplify the integrand. However, I think the mistake you are making is; you are forgetting to change the limits(1 & 1/4) of the integral. Whenever you make a variable change, the limits change according to the new variable.

Hope this helps...

 

[tronter]

multiply numerator and denominator by $$\sqrt{t}$$

You get: $$\int_{1}^{\frac{1}{4}} \frac{\cos \pi t}{t} \; dt$$

 

[trajan22]

A substitution should work, try u=pi(t)^.5

 

[Reshma]

You want
$$\int_{1}^{\frac{1}{4}} \frac{\cos \pi \sqrt t}{\sqrt t}dt$$

multiply numerator and denominator by $$\sqrt{t}$$

You get: $$\int_{1}^{\frac{1}{4}} \frac{\cos \pi t}{t} \; dt$$

Hey, that's wrong . Multiplying $\sqrt t$ in the numerator doesn't change $\cos \pi \sqrt t$ to $\cos \pi t$.

 

[stanners]

A substitution should work, try u=pi(t)^.5

Oh wow... haha, yeah.. thanks.

I was using u = cos pi(t)^.5

Bah, thanks.