What the calculator does
Choose one of three modes:
- Derivative. First to fifth derivatives, including partial derivatives when the expression has several variables.
- Indefinite integral. An antiderivative plus the constant C, with the method shown.
- Definite integral. The exact value from the Fundamental Theorem of Calculus where possible, otherwise an accurate numerical value, labelled as such.
How to enter functions
| Write | Meaning |
|---|---|
x^3, x^(1/2) |
powers |
2x, 3(x + 1), x sin(x) |
multiplication (the * sign is optional) |
sqrt(x), abs(x) |
square root, absolute value |
sin(x), cos(x), tan(x), sec(x), asin(x), atan(x) |
trigonometric functions, in radians |
e^x, exp(x), ln(x), log(x) |
exponential, natural log, log base 10 |
pi, e |
the constants π and e |
sin^2(x) |
(sin x)² |
Differentiation rules
The calculator applies these from the outside in. It names the rule for the outermost operation, differentiates each part, then simplifies.
Integration methods
- Standard integrals, such as ∫xⁿ dx = xⁿ⁺¹/(n + 1), ∫1/x dx = ln|x| and ∫eˣ dx = eˣ. These also apply to a linear inside function: ∫cos(3x) dx = sin(3x)/3.
- Substitution. If the integrand has the form g(u(x))·u′(x), set u = u(x) and integrate g(u) du.
- Integration by parts, ∫u dv = uv − ∫v du, for products like x·eˣ, x²·sin x, ln x and arctan x.
- Partial fractions for rational functions. Factor the denominator, split the fraction into simpler ones, and integrate each.
- Trigonometric identities, such as sin²x = (1 − cos 2x)/2, and rewriting odd powers of sin and cos.
The Fundamental Theorem of Calculus
If F′ = f and f is continuous on [a, b], then
The calculator first checks that the integrand has no singularity in the interval. If it does, the integral is improper and the theorem does not apply directly. Integrals like ∫ from −1 to 1 of 1/x dx are reported as improper rather than given a misleading value.
Worked examples
Derivative. For x² sin x, the product rule gives (x²)′ sin x + x² (sin x)′. The power rule and the derivative of sin then give
Indefinite integral. For ∫x eˣ dx, choose u = x and dv = eˣ dx, so du = dx and v = eˣ. Then ∫x eˣ dx = x eˣ − ∫eˣ dx, which gives
Definite integral. Since (arctan x)′ = 1/(1 + x²),
Common mistakes
- Forgetting the chain rule. The derivative of sin(3x) is 3cos(3x), not cos(3x).
- Dropping the absolute value. ∫1/x dx = ln|x| + C, which is valid for negative x as well.
- Forgetting + C on an indefinite integral.
- Using degrees. Calculus formulas for trigonometric functions assume radians.
- Applying F(b) − F(a) across a singularity. ∫ from −1 to 1 of 1/x² dx is not −2: the integrand is unbounded at 0.
Practical applications
- Physics. Velocity is the derivative of position, and displacement is the integral of velocity.
- Optimisation. Maximum and minimum values occur where the derivative is zero.
- Economics. Marginal cost and revenue are derivatives; totals are integrals.
- Probability. Probabilities for continuous distributions are areas under density curves.
- Engineering. Work, centres of mass and fluid pressure are computed with definite integrals.
Assumptions and limitations
Trigonometric functions use radians. Integrals are found with the methods listed above. Some integrals that need other techniques are not found, and those with no elementary antiderivative cannot be found by any method. In both cases the calculator says so, and a definite integral can still be computed numerically. Improper integrals, including those with infinite limits, are not evaluated.