AcademyIntegral Calculus
Academy
Antiderivatives
Level 1 - Calculus topic page in Integral Calculus.
Principle
An antiderivative reverses differentiation. If \(F'(x)=f(x)\), then \(F\) is an antiderivative of \(f\). A constant must be included because constants differentiate to zero.
Notation
\(\int f(x)\,dx\)
indefinite integral of f
\(F(x)\)
an antiderivative of f
\(C\)
constant of integration
The Core Method
Use reverse derivative rules.
Power antiderivative
\[\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\quad(n\ne-1)\]
Always include \(C\) for indefinite integrals. Check by differentiating the answer.
Worked Cases
Question
Find \(\int(6x^2-4x+5)\,dx\).
Answer
Integrate term-by-term. \(\int6x^2dx=2x^3\), \(\int-4x\,dx=-2x^2\), and \(\int5\,dx=5x\). Therefore the antiderivative is \(2x^3-2x^2+5x+C\).
Examples
Question
Verify that \(F(x)=\frac13x^3+7\) is an antiderivative of \(x^2\).
Answer
Differentiate \(F\): \(F'(x)=x^2+0=x^2\). Therefore \(F\) is an antiderivative of \(x^2\).