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The general antiderivative of $\cos x$ is $\sin x + C$, and the general antiderivative of $e^x$ is $e^x + C$. In both cases, $C$ represents an arbitrary constant, indicating that any vertical shift of these base functions is also a valid antiderivative.

Conditions: The variable is $x$.; The domain is all real numbers (or a connected interval thereof).

The general antiderivative of $\cos x$ is $\sin x + C$, and the general antiderivative of $e^x$ is $e^x + C$. In both cases, $C$ represents an arbitrary constant, indicating that any vertical shift of these base functions is also a valid antiderivative.

Conditions: The variable is $x$.; The domain is all real numbers (or a connected interval thereof).

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.

Treat the recurring integral as an algebraic variable (e.g., $I$). Move the term containing $I$ from the right-hand side to the left-hand side by adding it to both sides.

Conditions: The integral $I$ appears on both sides of the equation.; The coefficient of $I$ on the right side is $-1$ (or another constant).; Algebraic manipulation is permitted.

Treat the recurring integral as an algebraic variable (e.g., $I$). Move the term containing $I$ from the right-hand side to the left-hand side by adding it to both sides.

Conditions: The integral $I$ appears on both sides of the equation.; The coefficient of $I$ on the right side is $-1$ (or another constant).; Algebraic manipulation is permitted.