Integrals of exponential functions

The two rules

• Natural base: ∫ e^u du = e^u + C, so for a linear exponent ∫ e^ax+b dx = e^ax+b/a + C.
• Any positive base a ≠ 1: ∫ a^u du = a^u/ln a + C, so ∫ a^kx dx = a^kx/(k ln a) + C.
• Reason: d/dx(a^x) = a^x ln a, therefore the factor ln a must be divided out when integrating.
• If the exponent is a function g(x), the chain rule must be undone: ∫ e^g(x) g′(x) dx = e^g(x) + C.

Worked examples

1. ∫ e^{2x − 1} dx = e^{2x − 1}/2 + C.
2. ∫ 3^x dx = 3^x/ln 3 + C.
3. ∫ x e^x² dx: let u = x², du = 2x dx → ½∫ e^u du = e^x²/2 + C.
4. ∫ (e^x + 2^x) dx = e^x + 2^x/ln 2 + C — integrate term by term, different rules for each.