Antiderivatives and basic integrals
Running derivatives backwards
If you know a function's derivative, can you recover the function? Is there only one answer?
Every curve below has the same slope as every other at each x: they are all antiderivatives of the same function. Slide the constant C, or drag the curve through a point of your choosing, and watch which one is picked out.
The family y = F(x) + C, with the chosen curve in ink
y = f(x), the function whose antiderivatives these are
The indefinite integral
A function F is an antiderivative of f if F'=f. If F is one antiderivative, then every other one is F+C for some constant C, because two functions with the same derivative everywhere on an interval differ by a constant. The whole family is written as an indefinite integral:
Every rule comes from reading a derivative rule backwards. You can always check an answer by differentiating it.
| Integral | Result | Check |
|---|---|---|
| \int x^n\,dx,\ n\ne-1 | \frac{x^{n+1}}{n+1}+C | raise the power by 1, divide by the new power |
| \int \frac1x\,dx | \ln|x|+C | the one power the rule above misses |
| \int e^x\,dx | e^x+C | its own derivative |
| \int \cos x\,dx | \sin x+C | (\sin x)'=\cos x |
| \int \sin x\,dx | -\cos x+C | (-\cos x)'=\sin x |
| \int \sec^2x\,dx | \tan x+C | (\tan x)'=\sec^2x |
Term by term, and simplify first
Integrals split over sums and constant multiples: \int (af+bg)\,dx=a\int f\,dx+b\int g\,dx. There is no product or quotient rule for integrals, so rewrite products and quotients first. Expand (x+1)^2, divide \frac{x^2+1}{x} into x+\frac1x, and write roots as powers. Then integrate each term.
An extra condition, such as F(0)=2, picks one curve out of the family. That is what dragging the curve through a point does in the graph above.
Solve it step by step
Rewrite if needed, integrate term by term, then use the condition to find C. Fractions are fine. The score counts steps you answer correctly on the first try.
Common mistakes
- Forgetting + C
An indefinite integral is a whole family. Without +C, the answer names only one member.
- Integrating a product factor by factor
\int x\cdot x\,dx is \frac{x^3}{3}+C, not \frac{x^2}{2}\cdot\frac{x^2}{2}. Multiply out first.
- Using the power rule on 1/x
With n=-1, the power rule divides by 0. The antiderivative of \frac1x is \ln|x|.