Integral Calculator

Type a function, click Calculate. This integral calculator gives exact indefinite integrals instantly, right on the page.

• Function to integrate — the box comes pre-filled with x^2. Replace it with any function, or tap an example button such as sin(x), 1/x, 2x + 3, e^x, or cos(x)
• With respect to — pick the variable of integration: x, y, or t, matching the $dx$ (or $dy$, $dt$) in your problem

Click Calculate and the integral calculator returns the indefinite integral: for the pre-filled $x^2$ it gives $\dfrac{x^3}{3} + C$ — the whole family of functions whose derivative is $x^2$, constant included.

This page is more than a static integral 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 integral of 2x + 3?” The AI reads it and integrates it step by step.
• Take a photo of your homework. No typing: the AI reads the integrand and shows every step of the integration.
• Ask about the ideas, not just the answer. “Why does every integral end in + C?” The AI explains step by step — every result still comes from the integral calculator on this page, so nothing is guessed.
• Work backwards, then keep asking. Give an answer and ask what was integrated, or hand over a definite-integral word problem with bounds. Follow-up questions are welcome: ask “why?” as often as you need — the integral calculator page keeps teaching.

An integral reads two ways, and both are true at once.

Reading one: run differentiation backwards. You know a derivative — say $2x$ — and you ask which function it came from. $x^2$ works, because differentiating $x^2$ gives $2x$. That reverse question is integration, and its answer is called an antiderivative. This is the reading the integral calculator computes.

Reading two: accumulate area. The same symbol measures the area under a curve, stacked from infinitely many thin slices — each one $f(x)$ tall and $dx$ wide. Speed accumulates into distance; a rain rate accumulates into total rainfall.

The notation carries both readings:

$\int f(x)\,dx$

One symbol, one fact underneath: accumulating slices and reversing a derivative are the same operation. On this page the integral calculator works the reverse reading; ask the AI box below about areas when you need the other one.

The method you will reach for most, on this integral calculator and in every calculus course, is the power rule for integrals:

$\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C \quad (n\neq-1)$

Raise the exponent by one, divide by the new exponent, add C. It works because it is the derivative power rule played backwards: when you differentiate $\dfrac{x^{n+1}}{n+1}$, the $n+1$ in front cancels the one you divide by, and the exponent drops back to $n$.

Two companions make the rule cover whole expressions:

• The sum rule. The integral of a sum is the sum of the integrals — integrate $2x + 3$ one term at a time, as Example 2 below shows.
• The constant multiple. A coefficient rides along outside: $\int 2x\,dx = 2 \cdot \dfrac{x^2}{2} = x^2$.

One exponent is forbidden: $n = -1$. Raising it by one gives $x^0/0$, and division by zero stops the rule cold — $\dfrac{1}{x}$ integrates to $\ln\left|x\right| + C$ instead, a law unto itself. Beyond powers, the integral calculator also carries the standard shapes: $\sin x$, $\cos x$, and $e^x$ each integrate to a close cousin, worked in Example 2.

Evaluate $\int x^2\,dx$ — the function pre-filled in the integral calculator above.

1. Raise the exponent: $x^2$ becomes $x^3$.
2. Divide by the new exponent: $\dfrac{x^3}{3}$.
3. Add the constant:

$\int x^2\,dx=\dfrac{x^3}{3}+C$

Read the answer as a family, not a single curve: $\dfrac{x^3}{3} + C$ covers every upward and downward shift of one cubic, and every member of that family has slope $x^2$.

Now check — and make this a habit for life: an integral is correct exactly when differentiating it gives back the integrand.

$\dfrac{d}{dx}\left(\dfrac{x^3}{3}\right)=\dfrac{3x^2}{3}=x^2$

The derivative returns $x^2$ exactly, so the integral holds. No table lookup, no luck involved — the check is mechanical and always works. The integral calculator above confirms $\dfrac{x^3}{3} + C$ in one click; the AI box below will show the check line by line if you ask.

Two more integrations, both one click away in the integral calculator.

A trig integral. Type sin(x):

$\int \sin x\,dx=-\cos x+C$

Mind the minus sign — sine integrates to minus cosine. It is the single most-dropped sign in calculus, and it survives every shortcut: differentiate $-\cos x$ and you get $\sin x$ back, sign and all.

A sum, term by term. Type 2x + 3:

1. Split it: the sum rule turns one integral into two — $\int 2x\,dx$ and $\int 3\,dx$.
2. Integrate each: $2x$ gives $2 \cdot \dfrac{x^2}{2} = x^2$; the constant $3$ integrates to $3x$.
3. Recombine, one C for the whole answer:

$\int (2x+3)\,dx=x^2+3x+C$

Check by differentiating: $2x + 3$ comes straight back, one term per term. Both examples run live in the integral calculator above — type each one and compare the steps with what you get by hand.

Differentiation erases constants. The derivative of $x^2$, of $x^2 + 3$, and of $x^2 - 3$ is $2x$ every time, because a constant shifts a curve up or down without tilting it. So when the integral calculator runs the machinery backwards, the height is already lost — $\dfrac{x^3}{3}$, $\dfrac{x^3}{3} + 3$, and $\dfrac{x^3}{3} - 3$ all differentiate to $x^2$. No reverse step can choose among them, so the answer is the whole family at once, written with one + C.

A single known point picks the member you need: if the curve passes through a known value, substitute it and solve for C. That is how word problems turn a family into one curve.

Three slips the integral calculator sees every day:

1. Dropping the + C. Without it you report one curve out of infinitely many. Write it every time, even on quick work.
2. Using the power rule on $\dfrac{1}{x}$. $n = -1$ is the forbidden exponent — the answer is $\ln\left|x\right| + C$, not a power.
3. Forgetting to distribute over a sum. $\int(2x+3)\,dx$ integrates both terms: $x^2 + 3x + C$. Stopping at $x^2$ loses half the answer.