Finding Definite Integrals by Substitution and Solving Separable Differential Equations

Example 8: The technique is a little different for definite integrals. We can find new limits, and then we don’t have to substitute back. new limit We could have substituted back and used the original limits.

Example 8: Wrong! The limits don’t match! Using the original limits: Leave the limits out until you substitute back. This is usually more work than finding new limits

Example: (Exploration 2 in the book) Don’t forget to use the new limits.

Separable Differential Equations A separable differential equation can be expressed as the product of a function of x and a function of y. Example: Multiply both sides by dx and divide both sides by y 2 to separate the variables. (Assume y 2 is never zero.)

Separable Differential Equations A separable differential equation can be expressed as the product of a function of x and a function of y. Example: Combined constants of integration

Example 9: Separable differential equation Combined constants of integration

Example 9: We now have y as an implicit function of x. We can find y as an explicit function of x by taking the tangent of both sides. Notice that we can not factor out the constant C, because the distributive property does not work with tangent.

Until then, remember that half the AP exam and half the nation’s college professors do not allow calculators. In another generation or so, we might be able to use the calculator to find all integrals. You must practice finding integrals by hand until you are good at it! 