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Use integration by parts:

`int u(dv)/(dx) dx = uv - int v (du)/(dx) dx`

Letting `u = e^x`  and `v = 1/2sin2x`

`implies`  `int e^x cos2x dx = 1/2e^xsin2x - int 1/2e^xsin2x`

Using integration by parts for this last integral gives

`int 1/2e^xsin2x dx = -e^xcos2x - int -e^xcos2x dx`

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Use integration by parts:

`int u(dv)/(dx) dx = uv - int v (du)/(dx) dx`

Letting `u = e^x`  and `v = 1/2sin2x`

`implies`  `int e^x cos2x dx = 1/2e^xsin2x - int 1/2e^xsin2x`

Using integration by parts for this last integral gives

`int 1/2e^xsin2x dx = -e^xcos2x - int -e^xcos2x dx`

Putting these together we get

`int e^xcos2x dx = 1/2e^xsin2x - (-e^xcos2x +int e^x cos2x dx)`

`implies` `2 int e^xcos2x = 1/2e^xsin2x + e^xcos2x`

`implies` `int e^x cos2x = 1/4e^xsin2x + 1/2e^xcos2x` answer

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