Integration

Indefinite integration

For the direct function

Integration of the derivative gives the original function.

The integral of the sum gives the sum of the integrals.

The constant factor can be placed outside of the integral.

This formula reflects the changing variables rule in the integral.

This formula reflects the integration by parts rule.

This formula reflects the generalized integration by parts rule.

The integral of the sum is equal to the sum of the integrals of the summands (under some restrictions for convergence of the occurring infinite series).

The integral from the power series is equal to the sum of the integrals from each term of the series (inside some circle of convergence).

Repeated indefinite integration

This formula reflects repeated indefinite integration, where the integration in Mathematica should be performed with the option GenerateConditions->False: Integrate[(z-t)n-1(n-1)!f[t],{t,0,z}, GenerateConditionsFalse].

This formula reflects repeated indefinite integration, where the integration in Mathematica should be performed with the option GenerateConditions->False: Integrate[(Log[z]-Log[t])n-1t(n-1)!f[t],{t,0,z}, GenerateConditionsFalse].

Definite integration

For the direct function

This formula reflects the first mean value theorem.

This formula reflects the second mean value theorem.

This formula is called the Poincaré‐Bertrand formula.

Orthogonality

See Generalized Fourier series in the section Series representations.

Cauchy integrals

This formula is called the Cauchy‐type integral along the piecewise smooth contour (which can be closed or opened). The density function must be continuous along , but it must also meet a more stringent test known as the Hölder condition. The function satisfies the Hölder condition if, for two arbitrary points on the curve, for some positive constants and . The Cauchy‐type integral is analytic everywhere on the complex plane except on the contour itself, which is a singular line for this integral. Since the integral contains a factor (called the kernel) in the form of , it diverges at for any lying on .

This formula is called the Cauchy integral formula for the Cauchy integral. It is valid if is a closed, smooth contour enclosing the region on the complex plane, and the function is analytic over , continuous over , and represents the region outside of .

Singular integrals

In this Cauchy‐type integral, the singular point belongs to the contour and under the integrand function has a nonintegrable singularity. That is why this improper integral is called a singular integral. It can be evaluated, however, if a small neighborhood around this singularity is removed from the path of integration. The corresponding limit, as the size of the neighborhood shrinks to zero, is the Cauchy principal value of this divergent integral. For example:

In this example, represents the Cauchy principal value, and the contour is simply a straight segment on the real axis from to ; in other words, . Rather than integrating from through the point to the point b, you can integrate on the intervals and and then add these results to arrive at a value. By taking the limit of this calculation as , you can state the principal value.

This formula represents the Cauchy principal value of singular curvilinear integrals by the curve with a circular neighborhood , centered on and of radius ϱ, removed.

Sokhotskii formulas

General

This formula represents the function as a piecewise analytic function in the case when is a closed, smooth contour enclosing the region on the complex plane and represents the region outside of . If , the values Φ+(t0) and Φ-(t0) can be defined as the following limits:

If is an open contour with endpoints and , you can add an additional arbitrary curve segment connecting to (with the sense that corresponds to the limiting value from the left) and assign along this new segment. This extension allows you to apply the definitions for and to open contours .

These two formulas are called the Sokhotskii formulas. They were first derived by Y. V. Sokhotskii in 1873. Because they were later given a more rigorous treatment by J. Plemelj in 1908, they are often referred to as the Plemelj formulas. Sometimes the name Sokhotskii–Plemelj formulas is used.

The second of these formulas can be obtained from the Sokhotskii formulas by addition and subtraction.

In particular, if is analytical over , then and .

If the contour is a finite or infinite segment of the real axis, , these formulas hold for all , and so , . Thus is an analytic function with a jump discontinuity at , and the size of the jump is determined by the Sokhotskii formulas.

Example: The exponential integral Ei

After evaluation of this integral, you get:

For arbitrary , the Sokhotskii formulas give the following values:

It is important to note that for , the function is analytic and its limit values taken from either side of the real axis should agree with each other. This gives the relations:

This leads to the following behavior of :

Example: Beta-type integral

This integral can be called a beta‐type integral. It can be evaluated by the following formulas:

For arbitrary , the Sokhotskii formulas give the following results when you take into account the fact that, as , is continuous for but has a jump of size compared to when it is approached from the other side, .