Integration methods

Choose substitution or integration by parts to fit the structure and check the result.

Meaning

Substitution changes an expression's structure using a new variable. Integration by parts reverses the product rule. Consider both the function structure and the interval when choosing a method.

Example

In set so The result is Choose this substitution because the derivative of the inner x²+1 matches the factor 2x outside. Differentiating the result gives (x²+1)³×2x, checking the original integrand.A variable change also requires changing the differential. If a definite integral is evaluated entirely in the new variable, change both limits of integration as well.

Practice for this concept

Scope and references

Topics

Substitution, integration by parts, integrating various functions

Learning goal

Choose substitution or integration by parts to fit the structure and check the result.

Helpful background

Connect basic integrals with differentiation rules in reverse.