Question

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Answer 1

Answer:

perform integration by parts, until the original integral on the left hand side pops up in the right hand side.

Step-by-step explanation:

Set exdx=dv and cos(2x) =u

∫excos(2x)dx=excos(2x)−∫ex(−2sin(2x))dx

Now, here is the trick. Perform the integration by parts on the second term, keeping you u and v consist with what you did on your initial step.

Ie: exdx=dv and sin (2x)=u

∫exsin(2x)dx=exsin(2x)−∫ex(2cos(2x))dx

Plug this solution into your first solution, and then solve for ∫ex(2cos(2x))dx

∫excos(2x)dx=excos(2x)+2(exsin(2x)−∫ex(2cos(2x))dx)

∫excos(2x)dx=excos(2x)+2exsin(2x)−4∫excos(2x)dx)5

∫excos(2x)dx=ex5(cos(2x)+2sin(2x))