Free Online Integral Calculator

Name: Mathos AI
Rating: 4.5 (5000 reviews)

Integrate Faster, Learn the Steps

Stuck on integrals? Mathos AI solves them with free AI step-by-step explanations—just type your function or upload images to learn and verify your work.

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Step-by-step integral solutions

Our Integral Calculator explains the method, not just the answer—showing the antiderivative, applying u-substitution, integration by parts, or partial fractions when needed. For definite integrals, we evaluate with bounds using the Fundamental Theorem of Calculus: $\int_a^b f(x)\,dx = F(b)-F(a).$

AI-powered accuracy for complex integrals

Basic tools often fail on trickier expressions (nested functions, trig identities, exponentials, improper integrals, and double integrals). Mathos AI handles symbolic integration like $\int \frac{x}{x^2+1}\,dx$ and multivariable setups like $\iint_R (x^2+y^2)\,dA$ , while checking algebra and simplification along the way.

Type, paste, or upload a photo of your integral

Math notation is hard to type. With multimodal input, you can upload images of handwritten or textbook problems (e.g., $\int_0^{\pi} \sin(x)\,dx$ or $\int \sqrt{1-x^2}\,dx$ ) and get a readable integral plus a clear, guided solution.

What an integral is (and what your Integral Calculator returns)

An integral measures accumulation. In calculus, the most common meaning is area (net signed area) under a curve. The Integral Calculator typically returns either an indefinite integral (an antiderivative) or a definite integral (a number). For example, the indefinite integral $\int x^2\,dx = \frac{x^3}{3}+C$ returns a family of functions because many functions have the same derivative; the constant $C$ represents that vertical shift.

A definite integral includes bounds and produces a value: $\int_0^1 3x^2\,dx = \left[x^3\right]_0^1 = 1.$ Geometrically, this is the net area between $y=3x^2$ and the $x$ -axis from $x=0$ to $x=1$ . If the function dips below the axis, the integral counts that region as negative, which is why we call it signed area.

When you use an Integral Calculator with steps, you’re usually asking two things: (1) which integration technique applies (rules, substitution, parts, etc.), and (2) how to simplify the expression to a clean final result. Mathos AI focuses on both—helping you understand why a method fits, not just what buttons to press.

Definite vs. indefinite integrals: bounds, constants, and meaning

An indefinite integral solves for a function $F(x)$ such that $F'(x)=f(x)$ . That’s why results include +C. Example: $\int \cos(x)\,dx = \sin(x)+C.$ If your answer is missing $C$ , it’s incomplete in most symbolic-integration contexts.

A definite integral calculator evaluates $\int_a^b f(x)\,dx$ by finding an antiderivative $F$ and then applying the bounds: $\int_a^b f(x)\,dx = F(b)-F(a).$ This is the Fundamental Theorem of Calculus. For instance, $\int_{-1}^{2} (2x+1)\,dx = \left[x^2+x\right]_{-1}^{2} = (4+2)-(1-1)=6.$

Sometimes the bounds create special cases. With improper integrals, a bound may be infinite or the function may be undefined inside the interval. Then the integral is defined using a limit, such as $\int_1^{\infty} \frac{1}{x^2}\,dx = \lim_{b\to\infty}\int_1^b \frac{1}{x^2}\,dx.$ A step-by-step integral calculator should show that limit process clearly.

How to choose an integration method (rules, substitution, parts, partial fractions)

Choosing a method is the hardest part of “how to calculate integrals.” Start with pattern recognition. If you see a power of $x$ , use the power rule: $\int x^n\,dx = \frac{x^{n+1}}{n+1}+C\quad (n\ne -1).$ If you see $\frac{1}{x}$ , remember $\int \frac{1}{x}\,dx = \ln|x|+C.$ Trig and exponential basics include $\int e^x\,dx=e^x+C$ and $\int \sin(x)\,dx=-\cos(x)+C$ .

U-substitution (also called integration by substitution) works when you have a composite function and (almost) its derivative. Example: $\int 2x\cos(x^2)\,dx.$ Let $u=x^2$ , so $du=2x\,dx$ , giving $\int \cos(u)\,du = \sin(u)+C = \sin(x^2)+C.$ This is a classic “inside function + derivative” pattern.

Integration by parts is designed for products, based on $\int u\,dv = uv-\int v\,du.$ A common example is $\int x e^x\,dx.$ Choose $u=x$ and $dv=e^x\,dx$ to get $x e^x-\int e^x\,dx = x e^x-e^x+C = e^x(x-1)+C.$ For rational expressions like $\int \frac{2x+3}{x^2+x}\,dx$ , you may need algebraic simplification or partial fractions before integrating.

Beyond single-variable: double and triple integrals (multiple integration)

A double integral calculator evaluates integrals over a region in the plane: $\iint_R f(x,y)\,dA.$ This is used for area, mass, probability density, and more. If the region is a rectangle, you often compute it as an iterated integral: $\iint_R f(x,y)\,dA = \int_a^b\int_c^d f(x,y)\,dy\,dx.$ For example, $\int_0^1\int_0^2 (x+y)\,dy\,dx.$

A triple integral calculator extends this to 3D: $\iiint_E f(x,y,z)\,dV,$ useful for volume and density in space. Many problems become easier by switching coordinates (like polar, cylindrical, or spherical) when the region has symmetry. For instance, if a region is circular, polar coordinates can simplify the bounds and the integrand.

In multivariable contexts, the hardest parts are setting correct limits and including the correct area/volume element (like $dA$ or $dV$ ). A step-by-step integral calculator is especially helpful here because it can show the setup, not just the final number.

Frequently Asked Questions (FAQ)

How to calculate integrals?

To calculate integrals, use an Integral Calculator to identify an antiderivative or a technique like substitution or integration by parts. For definite integrals, compute $F(b)-F(a)$ after finding $F'(x)=f(x)$ .

What is the difference between definite and indefinite integrals?

An Integral Calculator returns an indefinite integral as an antiderivative with $+C$ , like $\int x\,dx=\frac{x^2}{2}+C$ . A definite integral includes bounds and returns a number, like $\int_0^1 x\,dx=\frac{1}{2}$ .

How do I do integration by parts?

An Integral Calculator uses integration by parts via $\int u\,dv = uv-\int v\,du$ . For example, $\int x e^x\,dx = x e^x-\int e^x\,dx = e^x(x-1)+C$ .

When should I use u-substitution?

Use an Integral Calculator with substitution when the integrand contains a composite function and its derivative, like $\int 2x\cos(x^2)\,dx$ . Let $u=x^2$ to get $\int \cos(u)\,du=\sin(u)+C$ .

What is an improper integral?

An Integral Calculator treats an improper integral as a limit when a bound is infinite or the function is undefined. Example: $\int_1^{\infty} \frac{1}{x^2}\,dx=\lim_{b\to\infty}\int_1^b \frac{1}{x^2}\,dx$ .

How do you solve a double integral?

A double integral calculator often converts $\iint_R f(x,y)\,dA$ into an iterated integral such as $\int_a^b\int_c^d f(x,y)\,dy\,dx$ . Then it integrates one variable at a time, keeping the other constant.