Most integration errors aren't conceptual — they're small input or setup slips that the maths then faithfully carries through. Here are the six most common integration mistakes, with a quick way to check each on the online integration calculator.
The six mistakes
1 · Degree mode for a trig integral
This is the number-one cause of "the calculator is broken" emails. Integral identities for sin, cos and tan only hold in radians.
Deg mode: ∫(sin(x), 0, π) gives a tiny, meaningless number.
Rad mode: ∫(sin(x), 0, π) → 2.
2 · Expecting total area, getting signed area
A definite integral is signed. Area below the axis subtracts, so symmetric functions can integrate to zero even though there's plenty of area.
∫(x^3, -1, 1) → 0 — the halves cancel.
Split at the root: ∫(x^3, 0, 1) → 0.25, then double to get 0.5.
3 · Swapping the limits
Order matters. Reversing the limits negates the result: ∫ab = −∫ba.
∫(x^2, 0, 3) → 9, but ∫(x^2, 3, 0) → −9. If your sign is off, check the order.
4 · Bracket / denominator slips
The calculator follows order of operations exactly — including your missing brackets.
1/x+1 is read as (1/x) + 1.
For 1/(x+1) type 1/(x+1). The same applies to roots and exponents: √(x+1), e^(2x).
5 · Integrating across a singularity
Numerical integration samples the function across the interval. If the function blows up inside the limits, the result is an error or nonsense.
∫(1/x, -1, 1) straddles the asymptote at x = 0 — undefined. Keep singular points out of the interval, or split around them.
6 · Using the wrong variable
The integrand must be written in x. Writing the function in t or n will not integrate as intended — rename it to x first. (Summation uses x as its index too — see evaluating summations.)
When unsure, test your setup on a case whose answer you already know (like ∫(x^2, 0, 3) = 9). If the known case is right, your syntax is right. This is the same habit recommended in the pillar guide.
Frequently asked questions
Why is my trig integral wrong?
Almost always degree mode — switch to Rad before integrating sin, cos or tan.
What happens if I swap the integration limits?
The sign flips: ∫ from a to b = −∫ from b to a.
Why does my integral error out or return a huge number?
There's likely a singularity inside the interval, such as 1/x at x = 0.
Check your work
Reproduce the "right" cases above and confirm your setup is sound.
Open the integration calculator →