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

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

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 new function then substitute back. Chain Rule

Integration using substitution Example Let u = 5x+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 = x2 + 1 du/dx = 2x du/2x = dx You end up with ….

Integration of Examples If you want to integrate a function such as When you do the substitution u = x3 - 1 du/dx = 3x2 dx = du/3x2 You end up with ….

Integration Examples continued General Case

Integration Examples continued