Verified answer

∫(e^x + cos(x))² dx

∫(e^(2x) + 2*e^(x)*cos(x) + cos²(x)) dx

∫e^(2x) dx + 2*∫e^(x)*cos(x) dx + ∫cos²(x) dx

e^(2x)/2 + 2*∫e^(x)*cos(x) dx + ∫cos²(x) dx

Integrate the second integral by integration by parts.

Integrate the third integral by using the half angle formula.