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