Core 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+f']. So integrating gives $e^x$*f(x).

Recognition strategy: When seeing $e^x$*(expression), try to split the expression into f + f' for some function f.

Common instances:

• $e^x$(sin x + cos x) = $e^x$[sin x + (sin x)'] => $e^x$ sin x + C
• $e^x$(1/x - 1/$x^2$) = $e^x$[$\frac{1}{x}$ + $\frac{1}{x}$'] => $e^x$/x + C
• $e^x$(x+1) = $e^x$[x + x'] => $xe^x$ + C
• $e^x$(tan x + $sec^2$ x) = $e^x$[tan x + (tan x)'] => $e^x$ tan x + C
• $e^x$(arctan x + $\frac{1}{1+x^2}$) = $e^x$[arctan x + (arctan x)'] => $e^x$ arctan x + C

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

Exponential-trig integrals:

• integral e^(ax)sin(bx) dx = e^(ax)(asin(bx)-bcos(bx))/(a^{2+b}^2) + C
• integral e^(ax)cos(bx) dx = e^(ax)(acos(bx)+bsin(bx))/(a^{2+b}^2) + C