Definite and Indefinite Integrals Without the Guesswork

Differentiation has rules you can follow mechanically. Integration has patterns you have to recognise, which is why a calculator that shows its steps is worth more here than for derivatives.

Indefinite versus definite

An indefinite integral asks which function differentiates to the one you typed. Integrate 3x² + 2x and you get x³ + x² + C. Check it by differentiating: 3x² + 2x, back where you started.

A definite integral asks for a number. Take the same function between 1 and 3: at x = 3 the antiderivative is 27 + 9 = 36, at x = 1 it is 1 + 1 = 2, so the answer is 34. The constant cancels in the subtraction, which is why definite answers never carry a C.

Reading the answer as area

The integral of x from 0 to 2 is 2, exactly the triangle with base 2 and height 2. Worth keeping as a sanity check.

Area below the axis counts as negative. The integral of sin x from 0 to 2 pi is 0, even though the curve clearly encloses space, because the hump above cancels the hump below. For geometric area, integrate the absolute value or split at the crossings and add the magnitudes; for sin x that gives 4.

The techniques the calculator is applying

Substitution

Take the integral of 2x cos(x²). Let u = x², so du = 2x dx and it collapses to the integral of cos u, which is sin u + C, or sin(x²) + C. Substitution is what a solver tries first, and spotting the inner function whose derivative is already in front of you is most of the skill.

Parts and partial fractions

Integration by parts handles products that substitution cannot. The integral of x e^x, taking u = x and dv = e^x dx, becomes x e^x minus the integral of e^x, giving x e^x - e^x + C.

Partial fractions handle rational functions. 1 / (x(x+1)) splits into 1/x - 1/(x+1), and each half integrates to a log, giving ln|x| - ln|x+1| + C. When an answer comes back full of logs, this is usually why.

When there is no closed form

Some integrals have no answer in elementary functions at all. The integral of e^(-x²) is the standard example, which is why the normal distribution is tabulated rather than solved. Calculators fall back on numerical methods.

Here is what that costs. The exact integral of 1/x from 1 to 4 is ln 4 = 1.3863. The trapezoid rule with three strips of width 1 gives 1.4583, about 5% high. Halve the strip width and the error falls to roughly a quarter of that, near 1.3%. Simpson's rule converges far faster on smooth curves, which is why most tools use it.

Using the integral calculator

You need three things: the expression, the variable of integration, and optionally the two limits. Leave the limits blank for an indefinite integral.

Syntax causes more wrong answers than the maths does. Write powers as x^2, and put an explicit asterisk in products, since 3x may not parse the way 3*x will. Use ln(x) when you mean the natural log, because many tools read log(x) as base 10. Exponentials are e^x or exp(x).

Parentheses are the other trap. 1/2x is ambiguous; type 1/(2*x) or (1/2)*x depending on what you meant. For improper integrals, most tools accept inf as a limit, so the integral of 1/x² from 1 to inf returns 1.

Check the result by differentiating it, which takes seconds in a free online maths tool. Plotting first helps too, since seeing where the curve crosses the axis tells you whether to expect cancellation; our roundup of free graphing calculators covers that.

Common questions

Why does my answer differ from the textbook by a constant? Both can be right. The integral of sin x cos x is either half sin²x + C or minus half cos²x + C, differing by exactly one half, which the constant absorbs.

Can I integrate across a discontinuity? Not straightforwardly. The integral of 1/x from -1 to 1 diverges, but some tools report 0 because the symmetry looks tidy. Split the range at the discontinuity and check each half separately.

What is the quickest way to check a definite integral? Estimate the area from a sketch. If the curve sits near a height of 2 across a width of 3 and the calculator returns 47, one of you has made a mistake.