Integration through substitution

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The integration by substitution or substitution rule is an important method in the integral calculus to primitive functions and definite integrals to calculate. By introducing a new integration variable, part of the integrand is replaced in order to simplify the integral and thus ultimately lead it back to a known or more easily manageable integral.

The chain rule from differential calculus is the basis of the substitution rule. Its equivalent for integrals over multidimensional functions is the transformation theorem , which, however, presupposes a bijective substitution function.

Statement of the substitution rule

Let be a real interval , a continuous function and continuously differentiable . Then


Let be an antiderivative of . According to the chain rule , the derivative of the composite function applies

By applying the main theorem of differential and integral calculus twice , one obtains the substitution rule:


We consider:

The goal is to simplify the partial term of the integrand to the integration variable. This is done by applying the substitution rule. To do this, you first multiply the integrand by and then, in a second step, replace the integration variable with everywhere . In a final step, the integration limits and are replaced by or .

So you educate

For the sake of clarity, in practice one often goes to a new integration variable via z. B. from to . Then the inverse function reads and the differential becomes from to and one obtains the formally equivalent expression:

Once you have found the antiderivative , you can evaluate it directly with the limits and or form the antiderivative to the original integrand as .

We can do the same backwards and apply the substitution rule

on. Then the integration variable has to be replaced by the term of and then multiplies the integrand by . Finally, one applies to the integration limits.

Substitution of a definite integral

example 1

Calculating the integral

for any real number : Through the substitution we get , therefore , and thus:


Example 2

Calculating the integral


The substitution gives , therefore , and thus


So it is replaced by and by . The lower limit of the integral is converted into and the upper limit into .

Example 3

For calculating the integral

you can , so substitute. From this it follows . With you get


The result can be with partial integration or with the trigonometric formula

and a further substitution. It turns out


Substitution of an indefinite integral

Requirements and procedure

Under the above conditions applies

After determining an antiderivative of the substituted function, the substitution is reversed and an antiderivative of the original function is obtained.

example 1

By completing the square and subsequent substitution , one obtains

Example 2

With the substitution one obtains

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

Special cases of substitution

Linear substitution

Integrals with linear chaining can be calculated as follows: If is an antiderivative of , then holds

if .

For example


there and .

Logarithmic integration

Integrals where the integrand is a fraction, the numerator of which is the derivative of the denominator, can be solved very easily with the help of logarithmic integration:


This corresponds to a special case of the substitution method with .

For example


since the derivative has.

Euler's substitution

According to Bernoulli's theorem, all integrals of the type


integrate elementary.


By substituting therefore , , and results


See also


  • Harro Heuser: Textbook of Analysis. Part 1 , 5th edition, BG Teubner, Stuttgart 1988, ISBN 3-519-42221-2 , p. 464
  • Konrad Königsberger: Analysis 1 , Springer, Berlin 1992, ISBN 3-540-55116-6 , pp. 200-201

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