Need help setting up for integration: (12sin2t+?56sin4t)^2

[MAC-A-TAC]

Would someone walk me through the process of setting up the following for integration....please. t = time

(12 sin 2t + ?56 sin 4t)[sup:30y78ihl]2[/sup:30y78ihl] NOTE: The only term under the radical is 56. Sorry don't have any math syntax software.

Thank you.

 

[soroban]

Re: Need help setting up for integration

Hello, MAC-A-TAC!

This is not a pleasant problem . . .

Would someone walk me through the process of setting up the following for integration?

. . \(\displaystyle \int (12\sin 2t + \sqrt{56}\sin 4t)^2\,dt\)

\(\displaystyle \text{We have: }\12\sin2t + 2\sqrt{14}\sin4t)^2 \;=\;\bigg[2(6\sin2t + \sqrt{14}\sin4t)\bigg]^2 \;=\;4\bigg[6\sin2t + \sqrt{14}\sin4t\bigg]^2\)

. . \(\displaystyle = \;4\bigg[36\sin^2\!2t + 12\sqrt{14}\sin2t\sin4t + 14\sin^2\!4t\bigg]\)

. . \(\displaystyle = \;4\left[36\left(\frac{1-\cos4t}{2}\right) + 12\sqrt{14}\sin2t(2\sin2t\cos2t) + 14\left(\frac{1-\cos4t}{2}\right)\right]\)

. . \(\displaystyle = \;4\bigg[18 - 18\cos4t + 24\sqrt{14}\sin^2\!2t\cos2t + 7 - \cos8t\bigg]\)

. . \(\displaystyle =\;4\bigg[25 - 18\cos4t - 7\cos8t + 24\sqrt{14}\sin^2\!2t\cos2t\bigg]\)


\(\displaystyle \text{We must integrate: }\:100\!\!\int\! dt \;\;-\;\; 72\!\!\int\!\cos4t\,dt \;\;-\;\; 28\!\!\int\!\cos8t\,dt \;\;+\;\; 96\sqrt{14}\!\!\int\!\sin^2\!2t\cos2t\,dt\)

. . \(\displaystyle \text{For the last integral, let: }\,u \:=\:\sin2t\)

 

[MAC-A-TAC]

Re: Need help setting up for integration

Thank you very much.