Formula: integral e^x[f(x) + f'(x)] dx = e^x * f(x) + C

Proof: d/dx[e^x * f(x)] = e^x * f(x) + e^x * f'(x) = e^x[f(x) + f'(x)].

Recognition pattern: When the integrand is e^x times (something), check if that something can be split into f + f'.

Examples:

• integral e^x(sin x + cos x) dx = e^x sin x + C [f = sin x, f' = cos x]
• integral e^x(1/x + (-1/x^2)) dx = integral e^x(1/x - 1/x^2) dx = e^x/x + C [f = 1/x]
• integral e^x(x+1) dx = integral e^x(x + 1) dx = e^x * x + C [f = x, f' = 1]
• integral e^x((x-1)/x^2) dx = integral e^x(1/x - 1/x^2) dx = e^x/x + C

Generalization: integral e^(ax)[af(x) + f'(x)] dx = e^(ax) * f(x) + C.