Antiderivative — with a detailed breakdown of every step and a graph
Function f(x)
Antiderivative
The antiderivative F(x) is a function whose derivative gives back the original: F′(x) = f(x). Any constant C can be added to an antiderivative, which is why it is always written.
How it was calculated — step by step
Every step names the technique used and shows the substitution, so the solution can be copied into your notebook.
Graph of the function and its antiderivative
The blue curve is the function f(x), the orange one is the antiderivative F(x). Check the link: where f(x) is positive the antiderivative increases; where f(x) is negative it decreases; where f(x) = 0 the antiderivative has a peak or a valley. This is exactly the inverse relationship compared with the derivative.
Table of antiderivatives
The core formulas. C is an arbitrary constant. Any row is easy to verify: differentiate the right column and you get the left one.
| f(x) | ∫f(x) dx |
|---|---|
| 0 | C |
| 1 | x + C |
| xⁿ, n ≠ −1 | xⁿ⁺¹/(n + 1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| aˣ | aˣ/ln a + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| 1/cos²x | tg x + C |
| 1/sin²x | −ctg x + C |
| 1/(1 + x²) | arctg x + C |
| 1/√(1 − x²) | arcsin x + C |
Definite integral — the area under the graph → · The inverse operation — the derivative →
📚 Theory: integration techniques
Integration is the operation inverse to differentiation. To find an antiderivative F for f means to pick a function whose derivative equals f. The check is always the same: differentiate the answer.
Linearity: the integral of a sum equals the sum of integrals, a constant factor is taken outside. Linear substitution: ∫f(kx + b) dx = F(kx + b)/k — when a linear expression stands under the integral instead of x.
Integration by parts: ∫u dv = u·v − ∫v du. It is applied to products like x·eˣ or x·sin x. Not every integral can be expressed through elementary functions — for example, ∫e^(−x²) dx has no closed formula, and that is normal.