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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).

To verify an antiderivative, you differentiate the proposed result. If the derivative of the proposed function equals the original integrand, then the proposal is correct.

Conditions: The proposed function is differentiable.; The original integrand is defined on the domain of interest.

To verify an antiderivative, you differentiate the proposed result. If the derivative of the proposed function equals the original integrand, then the proposal is correct.

Conditions: The proposed function is differentiable.; The original integrand is defined on the domain of interest.

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.