Integral Formulas
derivative of integral with x upper limit
(a∫x f(t) dt)' = f(x) Note that a is the lower limit and x the upper limit. X must be the upper limit for this to work. If a function like u(x) is the upper limit, substitute u(x) in as (f(u(x))) and then apply the Chain Rule (take derivative of u(x)).
∫1/√(u) = ?
2√(u) + C
- b∫a f(x) dx = ?
a∫b f(x) dx
Mean Value Formula for Definite Integrals
a∫b f(x) dx = f(c) × (b-a)
integral of tanx
or - |cosx|
substitution for x that must be made to evaluate the definite integral ∫√(x²-a²)
x = asect
substitution for x that must be made to evaluate the definite integral ∫√(a²-x²)
x = asint
substitution for x that must be made to evaluate the definite integral ∫√(x²+a²)
x = atant
integration of a constant to a power of u
∫a^u du = (a^u)/(lna) + C
integration of difference of squares in denominator (with u second)
∫du/(a²-u²) = (1/2a) × ln|(a+u)÷(a-u)| + C
integration of sum of squares in denominator
∫du/(u²+a²) = (1/a) × arctan(u/a) + C
integration of difference of squares in denominator (with u first)
∫du/(u²-a²) = (1/2a) × ln|(u-a)÷(u+a)| + C
integration of square root of sum of squares in denominator
∫du/√{(a²+u²)} = ln(u + √{u²+a²}) + C
integration of square root of difference of squares in denominator (u second)
∫du/√{(a²-u²)} = arcsin(u/a) + C
integration of square root of difference of squares in denominator (u first)
∫du/√{(u²-a²)} = ln(u + √{u²-a²}) + C
Integration by parts major formula
∫u×dv = u×v - ∫v×du