Calculus AP

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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)

  • Calculus AP -

    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

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