Integration by substitution method

On the basis of the method is following simple feature of indefinite integral:

variable substitution in the integral

We express initial integration variable x in terms of new variable t and get the expression for dx. Then we substitute derived expression into initial integral. We assume, that new variable is fitted such as new integral has the simplier form than initial.

Example:

integration by variable substitution method

we introduce new variable by using the formula:

new variable

then calculate the expression dt:

differential of the new variable

after this, we substitute new expression into initial integral:

expression of the integral in terms of new variable

When the change of variable is done, we get more simple integral:

expression of the integral after variable changed

which easily can be integrated:

solution of the integral in terms of the new variable

after calculations is done, we should go back to the old variable:

solution of the integral in terms of the old variable

thus, finally we get:

answer to the integral

Derived result can always be checked by the differentiation:

verification of the integral solution

Next example:

sample of the integral solution by substitution method

continue:

sample of the integral solution by substitution method

Use our online integrals calculator which automatically determines and makes optimal variable substitution to calculate your integral with step by step solution.

Other useful links:

Derivatives table
Area between crossed curves online calculator

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