Question 1

What is the indefinite integral and why is there a constant C?

Accepted Answer

The indefinite integral of f is the family of functions F whose derivative is f. Since the derivative of a constant is zero, if F is an antiderivative then F + C is too, for any number C. That is why ∫ 2x dx = x² + C: x² + 1 and x² − 7 also have derivative 2x. The constant can only be pinned down with an extra condition, such as F(0) = 3.

Question 2

How does the calculator check that the answer is right?

Accepted Answer

It differentiates the antiderivative it found with the derivative tool and compares the result with the integrand at up to 14 fixed test points, accepting a relative difference of at most 1e-7. The check table shows x, f(x), F′(x) and the difference. When both expressions are identical after expanding, that is also stated. An answer that fails the check is not shown as a result.

Question 3

Which integration techniques does it show in the step by step?

Accepted Answer

Power rule, constant multiple and linearity; exponential and logarithm; the trigonometric table; u-substitution; integration by parts (with u picked by the LIATE order) and the cyclic case of eˣ·sin x; partial fractions with polynomial division, repeated factors and irreducible quadratic factors; powers of sin and cos; products of sines and cosines; tan²x; and radicals of quadratics by completing the square. General trigonometric substitutions and multiple integrals are not included.

Question 4

How do I choose u in integration by parts?

Accepted Answer

The LIATE rule of thumb orders function families: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Pick as u the one that comes first on that list; the rest becomes dv. In x·eˣ, u = x (algebraic) and dv = eˣ dx, which gives (x − 1)·eˣ + C. In ln x, u = ln x and dv = dx, which gives x·ln x − x + C. The calculator applies this order and, if the first choice does not work, tries the second.

Question 5

How do you compute a definite integral?

Accepted Answer

Using the Fundamental Theorem of Calculus: if F is an antiderivative of f, the integral from a to b is F(b) − F(a). The constant C cancels. Example: the integral from 0 to π of sin x dx = [−cos x] from 0 to π = 1 + 1 = 2. The tool shows F(a), F(b) and the difference, and confirms with an independent numerical integration (tanh-sinh quadrature), displaying the relative difference between the two values.

Question 6

What is an improper integral and how do I tell if it converges?

Accepted Answer

It is an integral with an infinite limit or with the function blowing up inside the interval. It converges if the limit of the antiderivative exists and is finite. The classic tests: the integral from 1 to ∞ of 1/xᵖ converges if and only if p > 1 (then it equals 1/(p − 1)); the integral from 0 to 1 of 1/xᵖ converges if and only if p < 1. The calculator finds the problem points, probes the limit of the antiderivative and answers "converges" or "diverges"; the integral from −1 to 1 of 1/x² dx diverges, even though F(1) − F(−1) naively gives −2.

Question 7

Why do some integrals have no answer, such as e^(−x²)?

Accepted Answer

Some elementary functions have antiderivatives that are not elementary; Liouville's theory (1835) shows when this happens, and e^(−x²) is the classic example. There the antiderivative is defined by a special function: ∫ e^(−x²) dx = (√π/2)·erf(x) + C, ∫ sin x / x dx = Si(x) + C, ∫ eˣ/x dx = Ei(x) + C. The calculator writes it that way when it recognizes the pattern. Otherwise it says it found no closed form and, for definite integrals, returns the numerical value (the Gaussian from −∞ to ∞ gives √π ≈ 1.7724539).

Question 8

How do I type the function? Does it accept sin, tan, roots and absolute value?

Accepted Answer

Yes. Use x^2 or x², 2x for 2·x, sqrt(x) or √x, |x| or abs(x), pi or π, e for Euler's number, exp(x) or e^x. Functions: sin, cos, tan, arcsin, arctan, sinh, cosh, ln, log (base 10), sec, csc and cot; "sin^2(x)" and "sin x" also work. A trailing "dx" is ignored. A decimal comma is accepted. Under "Options" you can treat log as the natural logarithm, a common convention in English-language calculus.

Question 9

My notebook answer is different. Who is right?

Accepted Answer

Probably both, because antiderivatives differ by a constant and can be written in several forms: sin²x/2 and −cos²x/2 differ by 1/2, and ln|x| + ln 2 equals ln|2x|. To decide, differentiate your answer: if you get the integrand, it is right. The check table does exactly that for the tool's answer; use it as a template for yours. Also watch for identity swaps such as sin 2x = 2 sin x cos x.

Question 10

What are the tool's limits? Does my data leave the browser?

Accepted Answer

Your data does not leave: everything runs in the browser, and the second symbolic engine is only downloaded if the rules do not solve the problem. Limits: one integration variable at a time (other letters become constants), no double or triple integrals, no general trigonometric substitution, and functions such as floor, ceiling, remainder, factorial, min and max are not integrated. Denominators that do not factor into degree 1 or 2 polynomials with rational coefficients (such as x⁴ + 1) may fall to the support engine, which sometimes fails.

x	f(x)	F′(x)	Difference
-2.26376	-0.23533751	-0.23533751	2.8e-17
0.565523	0.99552794	0.99552794	1.1e-16
4.44149	377.08727	377.08727	5.7e-14
-0.222645	-0.17820488	-0.17820488	0
-0.357476	-0.25003303	-0.25003303	0
0.460526	0.72989202	0.72989202	2.2e-16

x	f(x)	F′(x)	Difference
-2.26376	-0.23533751	-0.23533751	2.8e-17
0.565523	0.99552794	0.99552794	1.1e-16
4.44149	377.08727	377.08727	5.7e-14
-0.222645	-0.17820488	-0.17820488	0
-0.357476	-0.25003303	-0.25003303	0
0.460526	0.72989202	0.72989202	2.2e-16

Integral Calculator with Steps