Integration by substitution

Integration by substitution or substitution rule is an important method in the integral calculus to calculate antiderivatives and definite integrals. By introduction of a new integration variable part of the integrand is replaced in order to facilitate the integral and thus ultimately to a known or easily manageable integral.

The chain rule of differential calculus is the basis of the substitution rule. Your equivalent for multidimensional integrals over functions of the transform set, however, presupposes a bijective substitution function.

  • 4.1 Requirements and Procedure
  • 4.2 Example 1
  • 4.3 Example 2
  • 5.1 Logarithmic Integration
  • 5.2 Euler's substitution
  • 5.3 Linear substitution

Statement of the substitution rule

Be a real interval, a continuous function and continuously differentiable. Then

Evidence

Be an antiderivative of. According to the chain rule for which the composite function derivative

By two applications of the main theorem of the differential and integral calculus is obtained so that the substitution rule:

Substitution of a definite integral

Example 1

Calculation of the integral

For any real number: By substituting obtained and:

Example 2

Calculation of the integral

By substitution we obtain, respectively, and thus

It is thus replaced by and by. The lower limit of the integral is converted into the upper limit and in the.

Example 3

Calculation of the integral

It substituted. As a result. With one obtains

The result can with Partial integration or with the trigonometric formula

And an additional substitution can be calculated.

Substitution of an indefinite integral

Conditions and procedures

Applies under the above conditions

Once you have determined a primitive function of the substituted function, one makes the substitution reversed and receives an antiderivative of the original function.

Example 1

With the substitution we obtain

Example 2

With the substitution we obtain

Note that the substitution is only for or only for strictly monotonic.

Special cases of the substitution

Logarithmic integration

Integrals with the special form of the numerator of the integrand is derivative of the denominator can be solved with the help of the logarithmic integration very simple, which is a special case of the substitution method:

Euler substitution

By a theorem of Bernoulli all integrals of the type and elemental can be integrated.

Example:

By substituting words, and the result is:

Linear substitution

Integrals with linear chains can be calculated as follows:

For the definite integral applies accordingly:

Integration by parts
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