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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.

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.

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.