`int e^(4x)cos(2x)dx`

To solve, apply integration by parts `int u dv = u*v - int vdu` .

In the given integral, the let the u and dv be:

`u = e^(4x) `  

`dv = cos(2x)dx`

Then, take the derivative of u to get du. Also, take the integral of dv to get v.

`du = e^(4x)*4dx`

`du = 4e^(4x)dx`

`intdv = int cos(2x)dx`

`v = (sin(2x))/2`

Substituting them to the integration by parts formula yields

`int e^(4x)cos(2x)dx= e^(4x)*(sin(2x))/2 - int (sin(2x))/2 * 4e^(4x)dx`

`int e^(4x)cos(2x)dx= (e^(4x)sin(2x))/2 - int 2e^(4x)sin(2x)dx`

For the integral at the right side, apply integration by...

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