An online integration and derivative calculator lets you evaluate calculus expressions instantly without re-deriving antiderivatives by hand. This guide covers the three calculus operations our free scientific calculator performs — definite integration, differentiation at a point, and summation — with the precise syntax for each and examples you can reproduce as you read.
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Numerical, not symbolic
The first thing to understand is what kind of calculus calculator this is. Tools split into two families:
- Symbolic calculators return a formula — e.g. ∫ x² dx = x³/3 + C.
- Numerical calculators return a number for a specific definite integral or a derivative at a specific point — e.g. the area under x² from 0 to 3 is exactly 9.
This calculator is numerical, matching the physical Casio fx-991ES PLUS. That is a strength for homework checking and engineering work: you get a high-precision decimal answer immediately. It does mean you supply limits (for integrals) or an evaluation point (for derivatives) every time, rather than getting back an antiderivative.
Because results are numerical, you'll occasionally see something like 1.9999999998 where the exact answer is 2. That is normal floating-point rounding from the underlying algorithm — see accuracy tips below.
Computing a definite integral
The definite-integral template uses this syntax:
∫(f(x), lower, upper)Press the ∫ key to insert the template, type the integrand in terms of x, then enter the lower and upper limits separated by commas. Press = to evaluate.
∫(x^2, 0, 3) → 9
The area under y = x² between x = 0 and x = 3. By hand, x³/3 evaluated from 0 to 3 gives 27/3 = 9. ✔
∫(sin(x), 0, π) → 2
With the angle mode set to Rad, the area under one hump of the sine curve is exactly 2. In Deg mode you would get a meaningless result.
For the full step-by-step version with more cases, see how to compute a definite integral online.
Computing a derivative at a point
Differentiation is evaluated at a single x-value:
d/dx(f(x), a)Here a is the point at which you want the instantaneous rate of change (the slope of the tangent line).
d/dx(x^3, 2) → 12
The derivative of x³ is 3x²; at x = 2 that is 3·4 = 12. ✔
d/dx(sin(x), 0) → 1
In Rad mode, the derivative of sin x is cos x, and cos 0 = 1. The curve climbs at 45° as it crosses the origin.
The mechanics of what the calculator does under the hood — and why it's an approximation — are covered in numerical differentiation explained.
Evaluating a summation (Σ)
The summation template adds up f(x) over a range of integers:
Σ(f(x), start, end)Σ(x^2, 1, 10) → 385
That's 1² + 2² + … + 10². Matches the formula n(n+1)(2n+1)/6 = 10·11·21/6 = 385. ✔
Summation has its own dedicated walkthrough: evaluating summations (Σ) step by step.
Entering functions correctly
Most wrong answers come from input, not from the maths. Use this reference for the integrand or summand:
| You want | Type | Example |
|---|---|---|
| Power xⁿ | x^n | x^3 |
| Square root | √( or sqrt( | √(x) |
| Exponential eˣ | e^( | e^(x) |
| Natural log | ln( | ln(x) |
| Log base 10 | log( | log(x) |
| Log custom base | log(b, x) | log(2, x) |
| Trig (radians) | sin( cos( tan( | cos(x) |
| Constant π | π | ∫(sin(x),0,π) |
| Fraction | / | 1/x |
Forgetting to switch to Rad mode before integrating or differentiating a trig function. Calculus identities like (sin x)′ = cos x only hold in radians.
More worked examples
∫(1/x, 1, 2) → 0.6931… (= ln 2)
∫(e^(x), 0, 1) → 1.7182… (= e − 1)
d/dx(ln(x), 1) → 1 (since (ln x)′ = 1/x, and 1/1 = 1)
Σ(1/x, 1, 5) → 2.2833… (the 5th harmonic number)
Tips for accurate results
- Set the angle mode first. Rad for any calculus involving trig.
- Bracket carefully.
1/x+1means (1/x)+1, not 1/(x+1). Wrap denominators:1/(x+1). - Expect tiny rounding. A result of 1.99999999 is the algorithm's numerical estimate of 2 — round sensibly.
- Avoid singularities inside the limits. Integrating 1/x across 0 (e.g. from −1 to 1) is undefined; the numerical method may return an error or a misleading value.
- Check against a known case when in doubt — that's exactly what the worked examples above are for.
Frequently asked questions
Does the calculator do symbolic or numerical calculus?
It is a numerical engine: it returns a precise decimal value for a definite integral or a derivative at a point, not an algebraic antiderivative. This mirrors the physical Casio fx-991ES PLUS.
What syntax do I use for a definite integral?
Use ∫(f(x), lower, upper). For example ∫(x^2, 0, 3) returns 9.
How do I find a derivative at a point?
Use d/dx(f(x), a), where a is the x-value. d/dx(x^3, 2) returns 12.
Why is my trig integral giving a strange answer?
Trig calculus assumes radians. Switch the angle mode to Rad before integrating or differentiating sin, cos or tan.
Is the online integration and derivative calculator free?
Yes — it runs entirely in your browser with no sign-up, download or usage limits.
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