Integration by Substitution Section 6.2a

substitution method of integration. The New Method A change of variables can often turn an unfamiliar integral into one that we can evaluate… This method is called the substitution method of integration.

The New Method 1. Substitute u = g(x), du = g (x)dx Generally, this method is used when integrating a composite function, and the derivative of the inside function is also present in the integrand: 1. Substitute u = g(x), du = g (x)dx 2. Evaluate by finding an antiderivative F (u) of f (u) 3. Replace u by g(x)

Initial Practice Problems Evaluate Let  Then Substitute: (Substitute)

Initial Practice Problems Evaluate Let Then Substitute:

Initial Practice Problems Evaluate Let Then Substitute:

Initial Practice Problems Evaluate Let Then Substitute:

Substitution in Definite Integrals Instead of the last step we’ve been using (re-substitution???): Substitute , and integrate with respect to u from to .

Evaluating Definite Integrals Evaluate Let Then Also, notice: Substitute:

Two Possible Methods? Method 1 Evaluate Let Then Also, notice: Substitute:

Two Possible Methods? Method 2 Evaluate Let Then Substitute:

Guided Practice Evaluate the given integral.

Guided Practice Evaluate the given integral.

Guided Practice Evaluate the given integral.

Guided Practice Evaluate the given integral.

Guided Practice Evaluate the given integral.