Antiderivative Calculator

The formula comes pre-filled. Click Calculate. This antiderivative calculator starts with ∫ cos(x) dx already in the math box — the whole formula, ready to edit.

• One math box, shown as real math. What you type appears in the antiderivative calculator as fractions, roots, and exponents — no special syntax to remember
• One-tap swaps. The buttons under the box — sin(x), tan(x), cot(x), sqrt(x) and more — replace the whole formula in the antiderivative calculator with one click
• Edit inside the formula. Change cos to sin, or the x to any expression such as 2x or 4x^3 - 2x + 5. The ∫ and the dx stay where they are; this antiderivative calculator does the rest

Click Calculate and the antiderivative calculator returns your antiderivative: the function whose derivative equals the input. The full family adds + C, because any constant disappears when you differentiate.

This page is more than a static antiderivative calculator — it is also an AI tutor you can ask questions. The question box under the form takes full questions, photos of the problem, and follow-ups:

• Ask in plain words. "What is the antiderivative of cot x?" The AI reads it, fills the box, and runs this antiderivative calculator for you.
• Take a photo of your homework. No typing: the AI reads the integrand and fills the antiderivative calculator form.
• Ask about the ideas, not just the answer. "Why is the antiderivative of 1/x a natural log?" The AI explains step by step — every result still comes from the antiderivative calculator on this page, so nothing is guessed.
• Work backwards, then keep asking. Give a slope and one point, and the AI shows how the + C gets pinned down. Follow-up questions are welcome: ask "why?" as often as you need.

To check any answer, differentiate it: you must get back exactly the function you typed into the antiderivative calculator.

Differentiation takes x² down to 2x. The antiderivative runs the question the other way: you know the slope is 2x — which function has that slope?

x² works: its slope at every point is 2x. But so do x² + 3 and x² − 3. Adding a constant shifts a curve up or down without changing its slope, because the derivative of any constant is zero. The antiderivative calculator cannot see the shift, and that is exactly why every antiderivative calculator answer ends in + C. Every output is a family, not a single curve.

Read the picture: both parabolas have slope 2x at every x. Reversing the derivative recovers the shape of a curve — never its height.

Now the symbols. What this antiderivative calculator computes is written as:

$\int f(x)\,dx = F(x) + C$

Every antiderivative calculator answer is read off this one equation.

The antiderivative calculator leans on these core rules:

One warning before you feed products to the antiderivative calculator: there is no product rule for integrals. $\int x e^x\,dx$ is not $\dfrac{x^2}{2} \cdot e^x$. Products need their own techniques — ask the AI on this antiderivative calculator page which one applies.

Find the antiderivative of $\cos x$.

1. Match it to the table: cosine integrates to sine.
2. So $\int \cos x\,dx = \sin x + C$.
3. Check by differentiating: $\dfrac{d}{dx}(\sin x) = \cos x$. The check holds.

The math box of the antiderivative calculator above comes pre-filled with cos(x) — click Calculate and it confirms sin(x) instantly. The same swap covers the rest of the trig family: the antiderivative of sin is $-\cos x + C$, while the antiderivative of tan and of cot each needs one rewrite first. Ask the antiderivative calculator page AI to walk you through it.

Evaluate $\int (4x^3 - 2x + 5)\,dx$.

1. Split into three integrals — the sum rule lets the antiderivative calculator integrate term by term.
2. Power rule each term: $\dfrac{4x^4}{4} - \dfrac{2x^2}{2} + 5x$.
3. So $\int (4x^3 - 2x + 5)\,dx = x^4 - x^2 + 5x + C$.
4. Check by differentiating: $4x^3 - 2x + 5$. The answer holds.

Replace the formula with 4x^3 - 2x + 5 in the antiderivative calculator above and it returns the same result.

Slope information alone leaves the height free. One extra point pins it down.

The slope of a curve is $\dfrac{dy}{dx} = \sin x$, and the curve passes through $(0, 3)$. Find the curve.

1. Integrate: $y = -\cos x + C$.
2. Substitute the point: $3 = -\cos 0 + C = -1 + C$.
3. So $C = 4$, and the curve is $y = -\cos x + 4$.

That is the whole point of the + C: the antiderivative calculator gives the family, and one known value selects the member you need.

Three slips the antiderivative calculator sees every day:

1. Dropping the + C. Without it you report one curve out of an infinite family. Add 10, subtract 1954 — every one differentiates back to the same slope.
2. Using the power rule on 1/x. The rule needs $n \neq -1$, and $\dfrac{x^0}{0}$ is undefined anyway. The antiderivative of $\dfrac{1}{x}$ is $\ln\left|x\right| + C$; the antiderivative calculator handles the special case for you.
3. Integrating a product as a product of integrals. $\int x e^x\,dx$ is not $\dfrac{x^2}{2} \cdot e^x$ — differentiate that and the extra term $\dfrac{x^2 e^x}{2}$ appears. Products need by parts or substitution, not the antiderivative calculator table.