Calculus AP
posted by Vicky on .
hi again
im really need help
TextBook: James Stewart:Essential Calculus, page 311. Here the problem #27: First make a substitution and then use integration by parts to evaluate the integral.
Integral from sqrt(pi/2) TO sqrt(pi)of θ^3 cos(θ^2)dθ.
i did my problem: let t=θ^2 then dt= 2θdθ.
∫θ^3cos(θ^2)dθ = ∫1/2 t cos t dt
=1/2∫tcostdt > let u=t, dv=cost dtthen du = 1, v= sin t
=1/2[tsint∫sint dt]
=1/2 tsint + 1/2 cos t + c.. reminder t=θ^2
=1/2 θ^2 sin(θ^2)+ 1/2 cos(θ^2) + c
so i got stuck and don't know how to solve with sqrt(pi) and sqrt(pi/2)

Up to 1/2∫tcostdt, you're OK.
After that, you need to integrate by parts.
I=(1/2)∫t cos(t)dt
=(1/2)[t sin(t)  ∫sin(t)]
=(1/2)[t sin(t) + cos(t)
To evaluate the definite integral, remember to adjust the limits accordingly, i.e.
from x^2 to t
sqrt(π/2)^2=π/2
sqrt(π)^2=π
I get (π+2)/4