Calculus Integration

Antiderivatives

Q: What is an antiderivative?

A function F whose derivative is the given function f; that is, F prime equals f. Antidifferentiation reverses differentiation.

Q: What is the reverse power rule?

The integral of x to the n is x to the n plus one over n plus one plus C, for n not equal to negative one: add one to the exponent and divide by the new exponent.

Q: How do you find the constant of integration?

Use an initial condition. Integrate to get F of x plus C, then substitute the known point and solve for C.

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Antiderivatives

California Calculus • Standard 15.0 • Integration

Antiderivatives is the opening topic of Integration in the California Calculus Standards. It is aligned to Standard 15.0, which requires students to demonstrate knowledge of the Fundamental Theorem of Calculus, beginning with antiderivatives and the indefinite integral.

An antiderivative of \(f\) is a function \(F\) with \(F'=f\). The indefinite integral \(\int f(x)\,dx=F(x)+C\) collects all of them, differing only by the constant of integration \(C\).

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Theory

An antiderivative of \(f\) is a function \(F\) whose derivative is \(f\). Because a constant differentiates to zero, every function has infinitely many antiderivatives that differ by a constant — captured by the +C in the indefinite integral.

\(F\) is an antiderivative of \(f\) when \(F'(x)=f(x)\). Antidifferentiation reverses the derivative.

The indefinite integral collects all antiderivatives at once:

\[\int f(x)\,dx=F(x)+C,\quad\text{where } F'(x)=f(x).\]

The constant of integration \(+C\) is essential: since \(\dfrac{d}{dx}(F+C)=f\) for any constant, the antiderivative is a whole family of parallel curves.

Key idea (reverse power rule): \(\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C\) for \(n\neq -1\). Add one to the exponent and divide by the new exponent.

Antiderivatives differ only by the constant \(C\).

An antiderivative \(F\) (navy) of a function \(f\) (teal).

The reverse power rule and the basic building blocks:

\[\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C\ \ (n\neq -1)\]

integral of x to the n is x to the n plus one over n plus one plus C

\[\int \dfrac{1}{x}\,dx=\ln|x|+C,\qquad\int e^{x}\,dx=e^{x}+C\]

integral of one over x is natural log; integral of e to the x is e to the x

Never forget \(+C\). An indefinite integral without the constant of integration is incomplete.

How to find an antiderivative

Rewrite roots and reciprocals as powers so the power rule applies.
Integrate term by term, adding one to each exponent and dividing.
Add \(+C\) — and if given a point, solve for \(C\).

Example 1 — Reverse the power rule

Find \(\displaystyle\int x^2\,dx\).

Solution

Add one to the exponent, then divide by the new exponent; include \(+C\).

\(\int x^2\,dx\)	\(=\)	\(\dfrac{x^{2+1}}{2+1}+C\)
	\(=\)	\(\dfrac{x^{3}}{3}+C\)

integral of x squared is x cubed over 3 plus C

Example 2 — Term by term

Find \(\displaystyle\int (3x^2-4x)\,dx\).

Solution

Integrate each term with the power rule, keeping the coefficients.

\(\int (3x^2-4x)\,dx\)	\(=\)	\(3\cdot\dfrac{x^3}{3}-4\cdot\dfrac{x^2}{2}+C\)
	\(=\)	\(x^3-2x^2+C\)

integral is x cubed minus 2 x squared plus C

Example 3 — A square root

Find \(\displaystyle\int \sqrt{x}\,dx\).

Solution

Write \(\sqrt{x}=x^{1/2}\); the new exponent is \(\dfrac{3}{2}\).

\(\int x^{1/2}\,dx\)	\(=\)	\(\dfrac{x^{3/2}}{3/2}+C\)
	\(=\)	\(\dfrac{2}{3}x^{3/2}+C\)

integral of root x is two thirds x to the three halves plus C

Example 4 — Find \(C\) from a point

Find \(f\) if \(f'(x)=2x\) and \(f(0)=5\).

Solution

Integrate, then use the initial condition to find \(C\).

\(f(x)\)	\(=\)	\(x^2+C\)
\(f(0)=5\)	\(\Rightarrow\)	\(C=5\)

So \(f(x)=x^2+5\).

f of x equals x squared plus 5

Common pitfalls

Do not forget \(+C\). Every indefinite integral needs the constant of integration.

The power rule fails at \(n=-1\). \(\int \dfrac{1}{x}\,dx=\ln|x|+C\), not \(\dfrac{x^0}{0}\).

Divide, do not multiply. Integrating raises the exponent and divides by the new exponent.

Frequently asked questions

What is an antiderivative?

A function \(F\) whose derivative is the given function \(f\); that is, \(F'=f\). Antidifferentiation reverses differentiation.

Why do you add + C?

Because the derivative of any constant is zero, every antiderivative is only determined up to a constant. The \(+C\) represents that whole family.

What is the reverse power rule?

\(\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C\) for \(n\neq -1\): add one to the exponent and divide by the new exponent.

How do you find the constant of integration?

Use an initial condition. Integrate to get \(F(x)+C\), then substitute the known point and solve for \(C\).

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Definite integrals (Riemann sums)

Antiderivatives

Antiderivatives

📖 Theory

How to find an antiderivative

Common pitfalls

Frequently asked questions

Theory