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

Proof: d/dx[e^xf(x)] = e^xf(x) + e^xf'(x) = e^x[f+f']. So integrating gives e^xf(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[(1/x) + (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 + 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