a) Additivity A sum under the integral is integrated by the summands are integrated separately and then summed.
b) Factor Rule A constant factor a can be pulled out of the integral.
c) If F(u) is the antiderivative of f(u), then the following relationship holds for arbitrary constants a and b with a not equal to 0.
d) If f(x) is differentiable and f(x) not equal to 0, then the following relationship applies.
e) Integration by parts Are the functions u(x) and v(x) are differentiable, then the following relationship holds.
f) Substitution If the function f(z) is continuous and z = g(x) is differentiable, then the following relationship holds.
Due to the additivity of the summands can be integrated individually.
The constant factors are taken outside the integral.
With the basic integrals, the solution follows.
The function under the integral is a function of a linear function.
Formation of the primitive of f with the substitution u = ax + b.
Install in accordance with rule c yields the solution of the integral.
That means in the numerator is the derivative of the denominator.
Install in accordance with rule d yields the solution of the integral.
Is the first factor v', then the integral of the first factor is v.
The derivative of the second factor is u'.
Put in following rule e.
With the integral of I the solution of the integral follows.
This can be written also shorter by factoring out.
Substitution of the denominator as a function g(x).
The derivative of g(x) by dx is the relationship according to the differentials.
Dissolved, the term can be used in the integral.
Put in and integrated.
Repatriation of substitution yields the solution.
The existence of the integrals in each case provided. There are a, b, c ∈ ℜ constants.
a) Interchange of the limits of integration.
b) Factor Rule A constant factor a can be pulled outside the integral.
c) Sum Rule A sum under the definite integral can be integrated by means of the summands.
d) Decomposition of the definite integral in part integrals.
e) Mean Value Theorem If f is integrable and we have m ≤ f ≤ M, then there exists at least one number μ with m ≤ μ ≤ M and we have the following relationship.
f) General Mean Value Theorem If f and g are integrable and we have m ≤ f ≤ M and g ≥ 0 either always or g ≤ 0, then there exists at least one number μ with m ≤ μ ≤ M and it works as follows.
g) Second Mean Value Theorem If f is monotonic and bounded and g integrable, then there exists at least one number μ for which the following relationship is valid.
h) If a function f is continuous and differentiable, then applies
i) Main Theorem of the Differential and Integral Calculations. If the antiderivative F of f is known, then the definite integral of f is calculated as follows
j) Integration by parts Are the functions u (x) and v (x) is differentiable, then the following relationship holds.
k) Substitution for definite integrals.