Every indefinite integral includes + C, the constant of integration. Since the derivative of any constant is zero, an infinite family of functions share the same derivative. C represents that unknown constant.
| Integral | Result | Condition |
|---|---|---|
| ∫ a dx | ax + C | a = constant |
| ∫ (1/x) dx | ln|x| + C | x ≠ 0 |
| ∫ eˣ dx | eˣ + C | Always |
| ∫ aˣ dx | aˣ / ln a + C | a > 0, a ≠ 1 |
| ∫ ln x dx | x ln x − x + C | x > 0 |
Evaluate: ∫₁³ (2/x) dx
∫ 2/x dx = 2 ln|x| + C[2 ln|x|]₁³ = 2 ln 3 − 2 ln 1= 2 ln 3This is derived using integration by parts: ∫ u dv = uv − ∫ v du
Then du = (1/x)dx, v = xx ln x − ∫ x · (1/x) dx = x ln x − ∫ 1 dx