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.