Antidifferentiation Up to this point in calculus, you have been concerned primarily with this problem: given a function, find its derivative Many important applications of calculus involve the inverse problem: given the derivative of a function, find the function
Antidifferentiation This operation of determining the original function from its derivative is the inverse operation of differentiation and is called antidifferentiation. Antidifferentiation is a process or operation that reverses differentiation.
Antidifferentiation What is the antiderivative of ? Notice: G is an antiderivative of F.
Antiderivatives & Indefinite Integrals Differential (variable of integration) Integral sign Integrand Antiderivative The antidifferentiation process is also called integration. Indefinite Integral The derivative of F is f.
Antiderivatives & Indefinite Integrals Q: What function has the derivative ? A: ln x This absolute value prevents you from having to find the natural log of a negative number. The Power Rule for Integration
Antiderivatives & Indefinite Integrals What if we were to shift the graph up 1 unit?
Antiderivatives & Indefinite Integrals Do the slopes change?
Antiderivatives & Indefinite Integrals The slopes stay the same.
Antiderivatives & Indefinite Integrals If a function has an antiderivative, then it has an infinite number of antiderivates.
Antiderivatives & Indefinite Integrals To capture the fact that there are infinitely many antiderivatives we add a constant. Constant of Integration
Conclusion Antidifferentiation is a process or operation that reverses differentiation. The antidifferentiation process is also called integration. Similar to differentiation, integration has a variety of rules that we must remember, recall, and be able to use.