Integration rules

Reverse power rule \int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \ne -1 The reverse power rule: the integral of x^n is x^(n+1)/(n+1) + C for n ≠ −1. Formula, examples with roots and negative powers, and the ln|x| exception. Linearity of the integral \int \left[a\,f(x) + b\,g(x)\right]dx = a\int f(x)\,dx + b\int g(x)\,dx The integral linearity rules: integrate sums term by term and pull constants out. Formulas, examples and the mistakes to avoid with products and quotients. U-substitution \int f(g(x))\,g'(x)\,dx = \int f(u)\,du, \quad u = g(x) The u-substitution rule reverses the chain rule: ∫f(g(x))g′(x)dx = ∫f(u)du. Formula, how to choose u, definite integral limits and examples. Integration by parts \int u\,dv = uv - \int v\,du The integration by parts formula ∫u dv = uv − ∫v du, how LIATE picks u, the definite integral version and short examples with ln x, x·eˣ and x·cos x. Definite integral properties \int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx The properties of definite integrals: reversing limits, zero width, additivity over intervals, odd and even symmetry and comparison bounds, with examples.