Integral Calculator

An integral calculator takes the grunt work out of one of calculus's most essential operations. Whether you're solving a textbook problem, checking your work, or just trying to remember how integration works, a good calculator walks you through the process fast. This page covers how to use the calculator, explains the core concepts behind integrals, and lays out the rules and techniques you'll run into most often. Bookmark it. You'll probably come back.

Enter Details

Definite integral of c·xⁿ from a to b.

a (lower)

b (upper)

Result

Enter c, n, and limits a and b to integrate c·xⁿ.

Uses ∫ c·xⁿ dx = c·xⁿ⁺¹/(n+1). Power n cannot be −1.

How to Use the Integral Calculator

Using the calculator is pretty straightforward. Enter your function using standard math notation, pick whether you want an indefinite or definite integral, and hit calculate. For definite integrals, you'll also need to enter a lower and upper limit.

A few tips that save headaches:

Use * for multiplication (write 3*x, not 3x)
Use ^ for exponents (x^2 means x squared)
Trig functions are usually entered as sin(x), cos(x), tan(x)
The natural log is typically ln(x), and Euler's number is e or exp(x)

If the calculator returns an error, double-check your parentheses. Mismatched brackets are the most common culprit. Some calculators also let you choose the variable of integration, which matters when your expression has multiple letters in it.

What Is an Integral?

At its simplest, an integral is the reverse of a derivative. Where differentiation breaks a function down into its rate of change, integration puts things back together. It answers questions like: if I know how fast something is changing, what was the original quantity?

There are two main types. An indefinite integral gives you a family of functions (plus a constant). A definite integral gives you a specific number, usually representing accumulated change or area over an interval.

Integration shows up everywhere: physics uses it to calculate displacement from velocity, economics uses it to find total cost from marginal cost, and engineering uses it to analyze signals and structures. It's not just a classroom exercise.

Indefinite Integral Calculator

An indefinite integral calculator finds the general antiderivative of a function. You plug in an expression, and it returns a new function whose derivative equals what you started with, plus a constant term.

For example, the indefinite integral of 2x is x² + C. The calculator handles the mechanics, but understanding what you're looking at matters when you need to interpret or use the result.

Antiderivatives and Integration

An antiderivative is any function F(x) such that F'(x) equals your original function f(x). Integration is the process of finding it. The two terms get used almost interchangeably in practice, though technically antidifferentiation is the operation and the result is the antiderivative.

Here's the thing: most functions have infinitely many antiderivatives, all differing by a constant. That's why the indefinite integral always includes + C at the end. The constant represents all those possible vertical shifts of the curve.

Not every function has a clean antiderivative. Some integrals can't be expressed in terms of standard functions at all, which is when numerical methods or special functions come into play. A calculator will tell you when it hits that wall.

Constant of Integration

The constant of integration, written as C, gets added to every indefinite integral result. It exists because when you differentiate a constant, it disappears. So when you work backwards, you have no way of knowing what constant (if any) was in the original function.

In practice, C only gets a specific value when you have extra information, like an initial condition. If you know that F(0) = 5, for instance, you can substitute that point into your antiderivative and solve for C. Without that kind of constraint, C just stays as a placeholder representing the whole family of solutions.

Definite Integral Calculator

A definite integral calculator evaluates an integral over a specific interval, from a lower limit a to an upper limit b. The output is a number, not a function. That number typically represents something concrete: area, total change, accumulated quantity.

The process relies on the Fundamental Theorem of Calculus, which connects differentiation and integration in a clean, powerful way. Essentially, you find the antiderivative and evaluate it at both limits, then subtract. The calculator does all of this automatically, but it's worth knowing what's happening underneath.

Evaluating Integrals Between Limits

To evaluate a definite integral from a to b, you compute F(b) minus F(a), where F is any antiderivative of your function. The constant of integration cancels out in this subtraction, which is why definite integrals give you a clean number instead of a + C.

A few things to watch for:

If the limits are reversed (b less than a), the result flips sign
If a function crosses zero within the interval, the integral can partially cancel itself out
Limits can be infinity, which leads to improper integrals that require special handling

The calculator manages these cases, but if you're getting a result that seems off, check whether your function has any discontinuities or sign changes inside your chosen interval.

Area Under the Curve

One of the most intuitive interpretations of a definite integral is area under the curve. If f(x) is positive over the interval [a, b], the definite integral gives you the exact area between the curve and the x-axis.

When the function dips below the x-axis, things get a little more nuanced. The integral assigns negative values to those regions, so they subtract from the total. If you want the total enclosed area regardless of sign, you'd need to split the integral at each zero crossing and take absolute values of each piece.

This geometric interpretation is genuinely useful. It connects abstract math to visual intuition, and it shows up constantly in probability (area under a density curve), physics (work done by a force), and statistics.

Integral Formulas and Rules

Knowing a handful of core formulas goes a long way. Most integrals you'll encounter in calculus courses, physics, or engineering can be solved by recognizing a pattern and applying the right rule. The calculator applies these automatically, but being familiar with them helps you set problems up correctly and catch mistakes.

The table below shows some common integral forms:

Function f(x)	Integral ∫f(x)dx
xⁿ (n ≠ −1)	xⁿ⁺¹ / (n+1) + C
1/x	ln\|x\| + C
eˣ	eˣ + C
sin(x)	−cos(x) + C
cos(x)	sin(x) + C
sec²(x)	tan(x) + C

These are the building blocks. More complex integrals are usually combinations or transformations of these forms.

Power Rule for Integration

The power rule for integration is probably the most-used formula in all of calculus. For any term of the form xⁿ where n is not equal to −1, the rule is:

∫xⁿ dx = xⁿ⁺¹ / (n+1) + C

You bump the exponent up by one, then divide by that new exponent. Simple. The rule works for fractions and negative exponents too, not just whole numbers. The one exception is x⁻¹ (or 1/x), which integrates to ln|x| + C instead, because the formula would otherwise require dividing by zero.

Polynomials are handled term by term using this rule, with constants factored out before integrating. So ∫(4x³ + 2x) dx becomes 4·(x⁴/4) + 2·(x²/2) + C, which simplifies to x⁴ + x² + C.

Exponential and Logarithmic Integrals

Exponential functions are particularly friendly to integrate. The integral of eˣ is just eˣ + C, no change. That's one of those math facts that seems too clean to be true, but it follows directly from the fact that eˣ is its own derivative.

For other exponential bases, like aˣ, the integral is aˣ / ln(a) + C. The natural log of the base shows up in the denominator, so you need a ≠ 1 for this to work.

Logarithmic integrals are a bit trickier. The integral of ln(x) is x·ln(x) − x + C, which you'd typically derive using integration by parts. The integral of 1/x is ln|x| + C, and that absolute value matters when x can be negative.

Trigonometric Integrals

Trig integrals are worth memorizing, at least the basic ones. They come up constantly, especially in physics and engineering problems involving waves, oscillations, and circular motion.

∫sin(x) dx = −cos(x) + C
∫cos(x) dx = sin(x) + C
∫tan(x) dx = −ln|cos(x)| + C
∫sec²(x) dx = tan(x) + C
∫csc²(x) dx = −cot(x) + C

More complicated trig integrals, like powers of sine and cosine or products of different trig functions, usually require identities or substitution to simplify before you can integrate. Knowing the Pythagorean identities and double-angle formulas helps a lot when you're doing those by hand.

Integration Techniques

When a function doesn't match a standard formula directly, you need a technique to transform it into something you can work with. There are several methods, and picking the right one is partly skill, partly pattern recognition.

Substitution (also called u-substitution) is usually the first thing to try. You replace a chunk of the integrand with a new variable to simplify the expression. It's essentially the chain rule in reverse.

Integration by parts handles products of functions, like x·eˣ or x·ln(x). The formula is ∫u dv = uv − ∫v du. Choosing which part to call u and which to call dv takes some practice.

Partial fractions work for rational functions (polynomials divided by polynomials). You decompose a complicated fraction into simpler pieces that are each easy to integrate.

Trigonometric substitution is useful when your integrand contains expressions like √(a² − x²) or √(x² + a²). You substitute a trig function for x to eliminate the square root.

Most calculus courses cover these in roughly this order. The calculator handles all of them, but working through a few by hand builds the intuition you need to set up problems correctly in the first place.