Integral Rules
Basic Integration Rules
| Rule | Formula |
|---|---|
| Constant Rule | ∫ c dx = cx + C |
| Power Rule | ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1) |
| ∫ x⁻¹ dx | ln|x| + C |
| Sum Rule | ∫ [f(x) ± g(x)] dx = ∫f dx ± ∫g dx |
| Constant Multiple | ∫ c·f(x) dx = c·∫f(x) dx |
| Integration by Parts | ∫ u dv = uv - ∫ v du |
| u-Substitution | ∫ f(g(x))·g'(x) dx = ∫ f(u) du, u=g(x) |
Common Integrals
| f(x) | ∫f(x) dx |
|---|---|
| eˣ | eˣ + C |
| aˣ | aˣ / ln(a) + C |
| sin(x) | -cos(x) + C |
| cos(x) | sin(x) + C |
| tan(x) | -ln|cos(x)| + C |
| sec²(x) | tan(x) + C |
| csc²(x) | -cot(x) + C |
| sec(x)tan(x) | sec(x) + C |
| 1/√(1-x²) | arcsin(x) + C |
| 1/(1+x²) | arctan(x) + C |
| 1/√(x²+a²) | ln(x + √(x²+a²)) + C |
| √(a²-x²) | (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C |