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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.

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).

Applying integration by parts twice returns the original integral $\int e^x \cos x dx$ on the right-hand side, but with a minus sign. This creates an algebraic equation where the unknown integral appears on both sides, allowing it to be solved by moving terms.

Conditions: The integrand is $e^x \cos x$.; Integration by parts is applied sequentially twice.; The cycle of derivatives for $e^x$ and $\cos x$ repeats.

Applying integration by parts twice returns the original integral $\int e^x \cos x dx$ on the right-hand side, but with a minus sign. This creates an algebraic equation where the unknown integral appears on both sides, allowing it to be solved by moving terms.

Conditions: The integrand is $e^x \cos x$.; Integration by parts is applied sequentially twice.; The cycle of derivatives for $e^x$ and $\cos x$ repeats.