Antiderivatives: Complete Lesson with Formulas and Methods

Definition

A function $F$ is an antiderivative of $f$ on an interval $I$ if:

$F'(x) = f(x) \text{ for all } x \in I$

Antidifferentiation is the inverse of differentiation.

Fundamental property

Two antiderivatives of the same function differ by a constant.

If $F$ and $G$ are antiderivatives of $f$, then $G(x) = F(x) + C$ for some $C \in \mathbb{R}$.

So we typically write “the” antiderivative as:

$F(x) + C$

Standard formulas

$f(x)$	Antiderivative $F(x)$	Domain
$k$ (constant)	$kx$	$\mathbb{R}$
$x^n$ ($n \neq -1$)	$\dfrac{x^{n+1}}{n+1}$	$\mathbb{R}$ (or $\mathbb{R}^*$ if $n < 0$)
$\dfrac{1}{x}$	$\ln\|x\|$	$x \neq 0$
$\dfrac{1}{\sqrt{x}}$	$2\sqrt{x}$	$x > 0$
$e^x$	$e^x$	$\mathbb{R}$
$\sin(x)$	$-\cos(x)$	$\mathbb{R}$
$\cos(x)$	$\sin(x)$	$\mathbb{R}$
$\dfrac{1}{1+x^2}$	$\arctan(x)$	$\mathbb{R}$

Linearity

$\int (af + bg) = a \int f + b \int g$

Antiderivative of a sum = sum of antiderivatives.

Example: antiderivative of $3x^2 - 5x + 7$? $F(x) = x^3 - \dfrac{5x^2}{2} + 7x + C$.

Chain pattern: $u' \cdot v'(u)$

If you spot $u'(x) \cdot v'(u(x))$, an antiderivative is $v(u(x))$.

Common cases:

• $\displaystyle \int u' \cdot u^n = \dfrac{u^{n+1}}{n+1}$ ($n \neq -1$)
• $\displaystyle \int \dfrac{u'}{u} = \ln|u|$
• $\displaystyle \int u' \cdot e^u = e^u$
• $\displaystyle \int u' \cdot \cos(u) = \sin(u)$

Example 1: antiderivative of $2x \cdot e^{x^2}$. $u = x^2$, $u' = 2x$. Antiderivative: $e^{x^2}$.

Example 2: antiderivative of $\dfrac{2x}{x^2 + 1}$. $u = x^2 + 1$. Antiderivative: $\ln(x^2 + 1)$.

Initial condition

Often we want THE antiderivative with a specific value, e.g. $F(0) = 3$.

Example: find $F$ antiderivative of $f(x) = 2x$ with $F(0) = 5$. $F(x) = x^2 + C$. $F(0) = C = 5$. So $F(x) = x^2 + 5$.

Connection to definite integral

$\int_a^b f(x)\, dx = F(b) - F(a)$

Fundamental theorem of calculus.

Common mistakes

1. Forgetting $+ C$.
2. Confusing antiderivative and derivative.
3. Not verifying: differentiate to check.
4. Dividing by $n + 1 = 0$: for $n = -1$ ($\dfrac{1}{x}$), antiderivative is $\ln|x|$, not $\dfrac{x^0}{0}$.

Solved exercises

Exercise 1

Antiderivative of $f(x) = 4x^3 - 6x + 2$.

Show solution

$F(x) = x^4 - 3x^2 + 2x + C$.

Exercise 2

Antiderivative of $\dfrac{3x^2}{x^3 + 1}$.

Show solution

$u = x^3 + 1$, $u' = 3x^2$. $F(x) = \ln|x^3 + 1| + C$.

Exercise 3

Antiderivative of $\cos(2x)$.

Show solution

$u = 2x$, $u' = 2$. $F(x) = \dfrac{1}{2}\sin(2x) + C$.

Takeaways

• $F' = f$ defines $F$.
• Antiderivatives differ by a constant.
• Always add $+ C$.
• Spot chain patterns $u' v'(u)$.
• Verify by differentiating.

Related

• Tips: Essential antiderivatives, Essential derivatives.
• Next: integration by parts, substitution.