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
To evaluate the integral from √(π/2) to √π of θ^3 cos(θ^2)dθ using the substitution method and then integration by parts, you made a good initial step by letting t = θ^2.
Now, you need to determine the new limits of integration corresponding to the new variable t.
For the lower limit √(π/2), substitute t = θ^2 to get t = (√(π/2))^2 = π/2.
For the upper limit √π, substitute t = θ^2 to get t = (√π)^2 = π.
So, the new limits of integration are from π/2 to π.
Now, let's go back to the integral:
∫[√(π/2) to √π] θ^3 cos(θ^2)dθ
Using the substitution t = θ^2, we have dt = 2θdθ. Rearranging, we get dθ = dt/(2θ).
Substituting these into the integral:
∫[π/2 to π] θ^3 cos(θ^2)dθ = ∫[π/2 to π] (1/2) t cos(t) dt
At this point, we can use integration by parts. Let u = t and dv = cos(t) dt. Then, du = dt and v = sin(t).
Using the integration by parts formula:
∫[π/2 to π] (1/2) t cos(t) dt = (1/2)(t sin(t) - ∫ sin(t) dt) from π/2 to π
Evaluating the indefinite integral of sin(t), we get ∫ sin(t) dt = -cos(t).
Substituting the limits of integration:
(1/2)[t sin(t) - (-cos(t))] from π/2 to π
Simplifying:
(1/2)[t sin(t) + cos(t)] from π/2 to π
Now, evaluate this expression at π and π/2, and subtract the value at π/2 from the value at π:
[(1/2)(π sin(π) + cos(π))] - [(1/2)(π/2 sin(π/2) + cos(π/2))]
Finally, simplify the expression to get the final result.