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The final antiderivative is $\frac{e^x(\sin x + \cos x)}{2} + C$. This result is obtained by solving the algebraic equation formed when the original integral reappears after two rounds of integration by parts.

Conditions: The function integrated is $e^x \cos x$.; The constant of integration $C$ is included.; The algebraic solution for the recurring integral is correct.

The final antiderivative is $\frac{e^x(\sin x + \cos x)}{2} + C$. This result is obtained by solving the algebraic equation formed when the original integral reappears after two rounds of integration by parts.

Conditions: The function integrated is $e^x \cos x$.; The constant of integration $C$ is included.; The algebraic solution for the recurring integral is correct.