Question

Integrate theta^2 cos(theta^2) 1/2 2 theta dtheta from sqrt(pi/2) to sqrt(pi)?

Answer 1

http://www.wolframalpha.com/input/?i=integrate+theta%5E2+cos%28theta%5E2%29+1%2F2+2+theta+dtheta+from+sqrt%28pi%2F2%29+to+sqrt%28pi%29

Answer 2

I have the solution but I'm wondering how to arrive to it.

I set x=theta^2
dx=2*theta d*theta and carried out integration by parts? But the thing is in the textbook solution, before integrating by parts, they remove the radical signs from the limits and I don't understand why. I'm supposed to do integration by parts and then take the limit from sqrt of so and so to so and so right? Why are the radicals/square roots removed in my textbook solution?

Is it alright if you can write up step by step how you arrived at your answer? Thanks a lot!

Answer 3

\[\int_{\sqrt{\frac\pi2}}^{\sqrt\pi}\frac12\theta^2\cos(\theta^2)(2\theta)d\theta=\int_{\sqrt{\frac\pi2}}^{\sqrt\pi}\theta^3\cos(\theta^2)d\theta\]for the substitution\[u=\theta^2\]\[du=2\theta d\theta\implies \theta d\theta=\frac{du}2\]if we change the bounds in terms of \(u=\theta^2\), we have\[\theta=\sqrt{\frac\pi2}\implies u=\frac\pi2\]\[\theta=\sqrt\pi\implies u=\pi\]so the integral necomes\[\frac12\int_{\frac\pi2}^{\pi}u\cos udu\]on which we perform integration by parts