Running the derivative backwards
In the last track, you started with a function f and found its derivative f' — the slope at every point. An antiderivative asks the opposite question: I hand you a slope function f, and you find a function F whose derivative is exactly f. In symbols, F is an antiderivative of f when F'(x) = f(x). You are not computing a new slope; you are searching for the original curve that would have produced this slope.
Here is a concrete picture. Suppose a car's speed (a rate of change of position) is a steady 60 km/h. Differentiation would have taken "position" down to "speed." Antidifferentiation climbs back up: knowing the speed, what was the position? The answer is 60t plus wherever you happened to start. That leftover starting point is the whole reason the answer isn't unique — and it leads straight to the next idea.
Why there are infinitely many: the + C family
Take f(x) = 2x. One antiderivative is F(x) = x^2, because the derivative of x^2 is 2x. But so is x^2 + 1, and x^2 + 5, and x^2 - 100. Every one of them differs only by a constant, and the derivative of any constant is 0 — so adding a constant changes nothing about the slope. That means there is never just one antiderivative: there is a whole family, written F(x) + C, where C is any constant.
Picture the graphs: x^2, x^2 + 1, x^2 + 5 are the same parabola shifted up and down, like a stack of identical curves. At any vertical slice, they all have exactly the same steepness — the same tangent slope — which is why they share the derivative 2x. The constant C just picks which curve in the stack you mean.
Notation and the basic reversals
Writing out "the family of all antiderivatives of f" is clumsy, so mathematicians invented a symbol: the indefinite integral, written integral f(x) dx = F(x) + C. The elongated S (integral) means "collect up," the dx says "with respect to the variable x," and the + C reminds you it's a family. Crucially, this is a family of functions, not a number — keep that contrast in mind for the next rung, where a definite integral with limits really will be a single number.
Because antidifferentiation is just differentiation run in reverse, every derivative rule you already know becomes an integral rule when you read it backwards. For powers, the power rule said "bring the exponent down and drop it by one." Backwards, you do the opposite: raise the exponent by one and divide by the new exponent.
integral x^n dx = x^(n+1) / (n+1) + C (for n != -1) integral 1/x dx = ln|x| + C (the n = -1 case) integral e^x dx = e^x + C integral cos x dx = sin x + C integral sin x dx = -cos x + C integral 1 dx = x + C
Three reversals, worked and checked
Let's actually use the kit. The beautiful thing about antiderivatives is that you can always check your own answer: just differentiate what you got and see if you land back on the original f. That self-check is your safety net — use it every single time.
- integral x^3 dx: raise the power to 4 and divide by 4, giving x^4 / 4 + C. Check: d/dx (x^4 / 4) = 4x^3 / 4 = x^3. Correct.
- integral (3x^2 + cos x) dx: handle each piece. The antiderivative of 3x^2 is x^3, and of cos x is sin x, so the answer is x^3 + sin x + C. Check: d/dx (x^3 + sin x) = 3x^2 + cos x. Correct.
- integral 5 dx: a constant 5 is just 5 times the antiderivative of 1, so the answer is 5x + C. Check: d/dx (5x) = 5. Correct.
Two habits carried over from differentiation make this painless. First, you can integrate a sum term by term — the integral of a sum is the sum of the integrals. Second, a constant multiplier slides straight through: integral of 3x^2 is 3 times the integral of x^2. With those two moves and the starter kit, you can reverse a surprising range of functions already.