Differentiation turns a function into its rate of change. Often we know the rate and want the function: a velocity, and we want the position. Running the rules from Constant, Power and Sum Rules, Derivatives of Trig Functions and Derivatives of Exponential and Log Functions backward gives antiderivatives.
Key idea
$F$ is an antiderivative of $f$ if $F'(x) = f(x)$. If $F$ is one antiderivative, every antiderivative has the form $F(x) + C$. We write $\displaystyle\int f(x)\,dx = F(x) + C$ (the indefinite integral). Power rule in reverse: $\displaystyle\int x^n\,dx = \frac{x^{n+1}}{n + 1} + C$ for $n \ne -1$.
Why it works
Why "$+ C$"? Adding a constant does not change a derivative, so $x^2$, $x^2 + 5$ and $x^2 - 3$ all have derivative $2x$. And there are no others: the The Mean Value Theorem shows that two functions with the same derivative on an interval differ by a constant. So "$+ C$" describes the whole family exactly. For the reverse power rule, differentiate the answer: $\frac{d}{dx}\frac{x^{n+1}}{n+1} = \frac{(n+1)x^n}{n+1} = x^n$.
| $f(x)$ | $\int f(x)\,dx$ |
|---|---|
| $x^n$ ($n \ne -1$) | $\frac{x^{n+1}}{n+1} + C$ |
| $\frac{1}{x}$ | $\ln\lvert x \rvert + C$ |
| $e^x$ | $e^x + C$ |
| $\cos x$ | $\sin x + C$ |
| $\sin x$ | $-\cos x + C$ |
| $\sec^2 x$ | $\tan x + C$ |
Worked example: A polynomial
Find $\displaystyle\int (3x^2 - 4x + 5)\,dx$.
- Term by term: $3\cdot\frac{x^3}{3} - 4\cdot\frac{x^2}{2} + 5x$.
- Add one constant: $x^3 - 2x^2 + 5x + C$.
- Check: the derivative is $3x^2 - 4x + 5$.
Answer: $x^3 - 2x^2 + 5x + C$
Worked example: Roots and fractions
Find $\displaystyle\int \left(\frac{2}{\sqrt{x}} + \frac{3}{x^2}\right)dx$.
- Rewrite: $2x^{-1/2} + 3x^{-2}$.
- Reverse power rule: $2\cdot\frac{x^{1/2}}{1/2} + 3\cdot\frac{x^{-1}}{-1} = 4x^{1/2} - 3x^{-1}$.
Answer: $4\sqrt{x} - \frac{3}{x} + C$
Worked example: Trig and exponential
Find $\displaystyle\int (\cos x + 2e^x - \sin x)\,dx$.
- $\int\cos x\,dx = \sin x$ and $\int 2e^x\,dx = 2e^x$.
- $\int(-\sin x)\,dx = \cos x$, because $\frac{d}{dx}\cos x = -\sin x$.
- One constant covers the whole answer.
Answer: $\sin x + 2e^x + \cos x + C$
Common mistake
Forgetting "$+ C$", or raising the power without dividing: $\int x^3\,dx = x^4$ is wrong because $\frac{d}{dx}x^4 = 4x^3$. Another trap: $\int x^{-1}\,dx$ would need division by 0, which is why it is $\ln\lvert x \rvert$. The universal check: differentiate your answer and see whether you get the integrand back.