Integral calculator with steps

This integral calculator works out indefinite and definite integrals with every step shown. Type a function and choose Definite or Indefinite. It gives the antiderivative plus C, or the exact value between two limits with the decimal alongside, and shades the area under the curve on a graph.

It opens on a definite integral solved by parts, with the area shaded. Every example below can be typed straight in, so you can compare your working with the calculator's, line by line.

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Integral Calculator

Integrate step by step and shade the area under the curve

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How to use the integral calculator

1. Choose Definite for an integral between two limits, or Indefinite for the antiderivative.
2. Type the function in the f(x) box. Leave out the integral sign and the dx.
3. For a definite integral, type the limits in From and To. A limit can be a number, a value like π/2, or infinity. You can also drag the two limit handles along the graph.
4. Read the answer, then the working. Each step says which rule it uses, such as the power rule, substitution or integration by parts.
5. For a definite integral, check the readouts: the numeric value, the antiderivative, and the area with every part counted as positive.

Any letter other than the variable is treated as a constant, so x and a in ax² are handled correctly, and the note under the answer says which letters it took as constants. To practise, open Practice and help for random definite integrals to give to 3 decimal places, with a hint and the full working for each.

How to type each function

The box turns what you type into maths as you go. Typing pi, sqrt or infty makes the symbol straight away.

Trig functions work in radians, so the limit for a right angle is π/2, not 90.

Indefinite integral calculator: the antiderivative plus C

An indefinite integral asks for the antiderivative: a function whose derivative is the one you started with. In Indefinite mode the calculator gives that function plus C, the constant of integration. The C is there because any constant disappears when you differentiate, so x³ + x² + 5 and x³ + x² − 2 have the same derivative.

Example 1: the power rule

Find the integral of 3x² + 2x. Integrate term by term: add one to each power, then divide by the new power.

Type 3x^2+2x in Indefinite mode and the calculator gives the same answer, with the steps named linearity and the power rule. To check any antiderivative, differentiate it: the derivative of x³ + x² is 3x² + 2x. Our Derivative Calculator does that check for you.

Definite integral calculator: the value between two limits

A definite integral gives a number. Find the antiderivative F, then subtract its value at the lower limit from its value at the upper limit. This is the fundamental theorem of calculus:

Example 2: a polynomial between two limits

Evaluate the integral of 6x² − 4x + 1 from 1 to 2.

The answer is 9. In Definite mode, type 6x^2-4x+1 with From 1 and To 2: the calculator gives 9 and the antiderivative 2x³ − 2x² + x + C. Now try one in the course, then press Mark to check it.

Example 3: area under the curve and negative area

In a definite integral, any part of the curve under the x-axis adds a negative amount. Evaluate the integral of x² − 4 from 0 to 3.

Here x is the variable and y is the value of x² − 4. The curve dips under the axis until x = 2, then rises above it. The piece under the axis has area 16/3 and the piece above has area 7/3, so the integral is 7/3 − 16/3 = −3. The total area, with both parts counted as positive, is 16/3 + 7/3 = 23/3, about 7.67.

The calculator shows both: −3 as the answer, and about 7.67 as Area, all counted positive. On its graph, the area above the axis and the area below are shaded in different colours. If a question asks for an area, split the integral where the curve crosses the axis, or use that readout.

Integral calculator with steps: the methods it uses

The working names each method as it goes, so you can see why it took each step. These examples all work in Indefinite mode.

For integration by parts it picks u with the LIATE order: logs first, then inverse trig, algebraic, trig and exponential. Every antiderivative it finds is checked by differentiating it back.

Example 4: integration by parts

Evaluate the integral of x cos x from 0 to π/2. Take u = x, so du = dx, and dv = cos x dx, so v = sin x. The rule is the integral of u dv equals uv minus the integral of v du:

The answer is π/2 − 1, about 0.5708. Type x cos(x) with From 0 and To pi/2 and the calculator shows the same. Its steps follow the same order: choose u by LIATE, put the parts into the rule, integrate sin, then subtract the value at 0 from the value at π/2.

Improper integral calculator: infinite limits

An improper integral has an infinite limit, or a function that blows up somewhere between the limits. The calculator handles both. It works out a limit at the bad end and says whether the integral converges to a number or diverges.

Example 5: does it converge?

Evaluate the integral of 1/x² from 1 to infinity. Replace infinity with b and let b grow:

It converges to 1. Type 1/x^2 with From 1 and To infty to check. Change the function to 1/x and the calculator reports that the integral diverges, because ln b grows without limit.

The last row is a trap. Putting the limits straight into ln|x| gives 0 − 0 = 0, which is wrong. The calculator spots that the function blows up inside the interval and says the fundamental theorem cannot be used straight across.

When there is no antiderivative

Some functions have no antiderivative made of standard functions. A famous one is e to the power −x², the bell curve behind the normal distribution. In Indefinite mode the calculator says No antiderivative found, rather than giving a wrong answer. Switch to Definite and it gives the value by numeric integration instead: from 0 to 1 it is about 0.74682, with a note saying where the number came from.

Double, triple and line integrals

This calculator does single integrals in one variable. It does not do double, triple or line integrals in one go. You can still use it for each layer of a double integral, working from the inside out: in Indefinite mode, other letters are treated as constants, so xy gives x²y/2 + C. Put in the inner limits by hand, then integrate the result with respect to the outer variable.

Common mistakes

• Forgetting the + C. Every indefinite integral needs it. Leaving it out usually costs a mark.
• Subtracting the wrong way round. It is F(upper) − F(lower), never the other way.
• Taking a negative integral as an area. Area below the axis counts as negative, so an integral can be smaller than the area it covers, or even 0.
• Integrating straight across an asymptote. If the function blows up between the limits, the integral is improper. Check for any x that makes a denominator 0.
• Forgetting to divide by the inside coefficient. The integral of e^(2x) is e^(2x)/2 + C, not e^(2x) + C.
• Using degrees. Calculus uses radians. A limit of 90 means 90 radians, not a right angle.

Practise integrals in a short timed paper

Integration runs through AP Calculus AB and AP Calculus BC practice, and through A-level Maths: Calculus in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics. To learn integration from the start, the Calculus course builds up from area under a curve to the fundamental theorem.

Read next

Maths · CalculusWhat is an integral? Meaning, area and worked examplesWhat an integral is: area under a curve, adding up tiny pieces, and the reverse of differentiation. Definite and indefinite integrals with worked examples.