Integration by Substitution

Integration by substitution, also known as u-substitution, is a technique used to simplify integrals by substituting a new variable for a part of the integrand. This method is based on the chain rule of differentiation and is particularly useful for integrals containing composite functions.

The basic idea is to make a substitution that transforms the integral into a simpler form. The formula for integration by substitution is:

Where and .

This technique is essentially the reverse of the chain rule for differentiation:

Integration by substitution is particularly useful in the following situations:

• When the integrand contains a composite function, such as
• When the integrand has a function and its derivative, like
• When the integrand can be rewritten in the form

The key to successfully applying this method is recognizing which part of the integrand should be substituted with a new variable. Look for:

• An "inner function" that appears as part of a composite function
• A function whose derivative (or a multiple of its derivative) also appears in the integrand

1. Identify the substitution: Choose a part of the integrand to substitute with a new variable u.
2. Find the differential: Calculate du by differentiating u with respect to x.
3. Express dx in terms of du: Solve for dx from the differential equation.
4. Substitute into the integral: Replace the chosen part with u and dx with the expression in terms of du.
5. Evaluate the new integral: Integrate with respect to u.
6. Back-substitute: Replace u with the original expression in terms of x.

Here are some common substitutions that are useful in various integration problems: