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$$\frac{(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.