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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.