Integration To integrate ex To be able to

Integration • To integrate ex • To be able to integrate functions where the numerator is the derivative of the denominator

Integration Remember § Differentiation and Integration are inverses § Multiply by power and reduce power by 1 § Add 1 to power and divide by new power

Differentiation if if kex f(x) = g(x) = eax x e f `(x) = x ke g`(x) = ax ae Chain rule: u = ax du/dx = a y = eu dy/du = eu dy/dx = eu x a = a eu = aeax Integration Chain Rule

Integration using substitution The Chain rule for differentiation provides a useful technique for integration In the chain rule we introduce a new variable u. We can do the same in integration We must also replace dx Integrate Chain Rule new function then substitute back.

Integration using substitution Example Let u = 5 x+3 then =5 To replace dx we can separate the variable to give du = 5 dx which gives We now have Which we can now integrate

Integration using substitution Example Integrate Now substitute back

Integration using substitution - special case Integrate functions where the numerator is the derivative of the denominator ie

Integration of Examples If you want to integrate a function such as or (note that the numerator is the denominator differentiated)

Integration of Examples If you want to integrate a function such as When you do the substitution u = x 2 + 1 du/dx = 2 x du/2 x = dx You end up with ….

Integration of Examples If you want to integrate a function such as When you do the substitution u = x 3 - 1 du/dx = 3 x 2 dx = du/3 x 2 You end up with ….

Integration Examples continued General Case

Integration Examples continued