- What's the difference between a definite and an indefinite integral?
- An indefinite integral is a family of functions: every F(x) whose derivative is your f(x). That's why the answer ends in + C, because adding any constant doesn't change the derivative. A definite integral is one number: the signed area under f(x) between two bounds, found as F(b) - F(a). The constant cancels in that subtraction, so definite answers have no + C. For x^2, the indefinite integral is (1/3)*x^3 + C, and the definite integral from 0 to 1 is 1/3.
- Why does every antiderivative end in + C?
- Because differentiation throws constants away. The derivative of x^3 is 3x^2, and so is the derivative of x^3 + 7 and x^3 - 100. When you run that backwards, you can't know which constant was there, so the answer is written as x^3 + C to mean all of them at once. Leaving off the + C on an indefinite integral is one of the most common ways to lose a point on a calculus test.
- How does the calculator decide which rule to use?
- It rewrites the function as a sum of terms first, then matches each term against a fixed rule set. A plain power of x gets the power rule. e^x, sin, cos, tan, and 1/cos(x)^2 come from a table. A function of a linear expression, like cos(3x) or (2x+1)^3, gets a substitution u = 3x or u = 2x + 1. A power of x multiplied by e^x, sin, cos, or ln(x) gets integration by parts. It's the same order a textbook teaches, which is why the steps read like worked solutions instead of a black box.
- When do you use integration by parts?
- When the integrand is a product where one piece gets simpler when you differentiate it and the other stays easy to integrate. The formula is ∫u dv = u*v - ∫v du. For x*e^x, take u = x (its derivative is 1) and dv = e^x dx (its integral is e^x), which gives x*e^x - ∫e^x dx = x*e^x - e^x + C. For x^2*sin(x) you have to apply it twice. The steps show the tabular shortcut: differentiate x^2 down to a constant, integrate sin(x) at each stage, and alternate the signs.
- Why does it say there's no closed form for e^(-x^2)?
- Because e^(-x^2), the bell curve behind the normal distribution, has no antiderivative made of ordinary functions. That's a proven result, not a gap in this tool. Mathematicians gave its integral a name, the error function erf(x), precisely because it can't be written any other way. Plenty of innocent-looking functions are like this, including sin(x)/x and sqrt(1 + x^3). Switch to Definite and you still get the number: from 0 to 1, e^(-x^2) integrates to about 0.746824.
- Why won't it integrate 1/x^2 from -1 to 1?
- Because 1/x^2 shoots to infinity at x = 0, which sits between the bounds, so the area is infinite. If you apply the antiderivative -1/x blindly, you get F(1) - F(-1) = -1 - 1 = -2, a negative area for a function that's never negative. That's the classic trap. This calculator scans the interval for points where the function stops being finite before it evaluates anything, and tells you there's no finite answer instead of printing -2.
- How accurate are the definite integral values?
- When there's a symbolic antiderivative, the value comes from F(b) - F(a), and it's exact whenever the answer is a fraction with a denominator up to 1,000 (you'll see 1/3 ≈ 0.333333). Every one of those answers is also recomputed by adaptive Simpson's rule, a numerical method that splits the interval into pieces until the estimate stops changing, and the result is only shown if the two agree. Numeric-only answers are computed with a tolerance of 1e-8 and shown to 6 significant digits.
- What does ln(abs(x)) mean in the integral of 1/x?
- It's ln|x|, written in plain text. The natural log ln(x) only exists for positive x, but 1/x is defined for negative x too. The absolute value makes the antiderivative work on both sides of zero: the derivative of ln|x| is 1/x whether x is 3 or -3. You'll see the same pattern with substitution, like (1/2)*ln(abs(2*x + 1)) for 1/(2x + 1).
- What can't this calculator do yet?
- It integrates functions of x only, one variable at a time, so no double or triple integrals and no bounds at infinity. Its rule set covers powers, exponentials, sin, cos, tan, sec squared, linear substitution, and integration by parts. It doesn't do trig substitution, partial fractions, or inverse-trig results like arctan(x), so 1/(x^2 + 1) gets the no-closed-form message in Indefinite mode. The definite value still works for all of those.
- Is what I type sent anywhere?
- No. The parser, the integrator, and the numeric check all run in your browser. Nothing is sent to a server, nothing is stored, and closing the tab clears it. That's also why results show up as you type instead of after a round trip.
