Pierre-Simon Laplace 1749 – 1827 Pierre-Simon Laplace 1749 – 1827 Laplace proved the stability of the solar system. In analysis Laplace introduced the potential function and Laplace coefficients. He also put the theory of mathematical probability on a sound footing.

The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for only a limited range of functions. We can sometimes use substitution to rewrite functions in a form that we can integrate.

Example 1: The variable of integration must match the variable in the expression. Don’t forget to substitute the value for u back into the problem!

Example 2: One of the clues that we look for is if we can find a function and its derivative in the integrand. Note that this only worked because of the 2x in the original integrand. Many integrals can not be done by substitution. The derivative of is. Separation of variables

Example 3: Solve for dx.

Example 4:

x f (x) y g (x) Since the integrand has a symbolic antiderivative, we can write g(x) as

Example 5: We solve for because we can find it in the integrand.

Example 6:

Example 7: For the definite integral, we need the FTC Part 2. new limit Must Change the original limits!

Example 8: Don’t forget to use the new limits. New Limits

Example 9: New Limits

Example 10: New Limits

Example 11: