Indefinite integrals of polynomials

What is an indefinite integral?

A function $F(x)$ whose derivative is $f(x)$ is called an antiderivative of $f(x)$. For example, $(x^3)' = 3x^2$, so $x^3$ is an antiderivative of $3x^2$.

Since $x^3 + 1$ and $x^3 - 5$ also differentiate to $3x^2$, there are infinitely many antiderivatives, differing by constants. Using an arbitrary constant $C$ we write

and call this the indefinite integral of $3x^2$. $C$ is the constant of integration. Always add $+C$ to an indefinite integral.

The basic formulas

So a polynomial can be integrated term by term: raise the power by one and divide by the new power. For a constant $a$, $\int a\,dx = ax + C$.

Example: $\displaystyle\int (6x^2 - 4x + 5)\,dx = 6 \cdot \frac{1}{3}x^3 - 4 \cdot \frac{1}{2}x^2 + 5x + C = 2x^3 - 2x^2 + 5x + C$

Expand products first

You cannot apply the formula directly to a product such as $\int (x+1)(x-3)\,dx$. Expand it into a polynomial first.

A power of the form $(x+a)^n$ can be integrated without expanding. The result looks different from the expanded answer only by a constant, so both are correct.

Finding a function from conditions

For a problem such as "find $f(x)$ with $f'(x) = 3x^2 - 2x$ and $f(1) = 3$", integrate first to get $f(x) = x^3 - x^2 + C$, then use $f(1) = 3$ to find $C$: $1 - 1 + C = 3$, so $C = 3$ and $f(x) = x^3 - x^2 + 3$.

"The slope of the tangent line to $y = f(x)$ at $(x,\ y)$ is ..." simply means $f'(x) = \ldots$. A point on the curve then fixes the constant.

Common mistakes

• Forgetting $+C$: an indefinite integral always needs the constant.
• Forgetting to divide: $\int x^2\,dx = x^3$ is wrong; it is $\frac{1}{3}x^3 + C$.
• Constant terms: $\int 5\,dx = 5x + C$, not $5$ or $0$.
• Integrating a product factor by factor: $\int (x+1)(x-3)\,dx$ is not $\int (x+1)\,dx \times \int (x-3)\,dx$. Expand first.