Advanced Antiderivative Calculator

x	F(x)
0	0.000000
1	1.000000
2	8.000000

Formula Used

Antiderivatives reverse differentiation. For a supported input family, the calculator applies a matching integration rule and adds an arbitrary constant.

Power rule: ∫a·xⁿ dx = a·xⁿ⁺¹/(n+1) + C, when n ≠ -1
Log rule: ∫1/x dx = ln|x| + C
Exponential rule: ∫a·e^bx dx = (a/b)e^bx + C
Trig rules: ∫sin(u)dx, ∫cos(u)dx, ∫sec²(u)dx, and related forms use substitution
Inverse trig rules: ∫1/(1+u²)dx and ∫1/√(1-u²)dx convert to arctan and arcsin forms

FAQs

1. What does this calculator return?

It returns a symbolic antiderivative for supported function families, sample output values, a graph, and downloadable CSV and PDF-friendly output.

2. Does it solve any arbitrary expression?

No. It is rule-based and designed for common antiderivative patterns such as powers, exponentials, logarithmic forms, trigonometric forms, and selected hyperbolic forms.

3. Why is there a constant of integration?

Every antiderivative represents a family of functions. Differentiating any added constant gives zero, so all valid antiderivatives differ by a constant.

4. What happens if the domain is restricted?

The calculator shows a domain reminder. Reciprocal, logarithmic, and inverse trigonometric forms may be undefined for some x values in the chosen range.

5. Why might the graph skip some points?

Undefined or non-real values are excluded from the graph data. This usually happens near asymptotes, singularities, or invalid square-root inputs.

6. Can I use this for checking homework?

Yes. It is useful for checking standard forms and understanding which integration rule applies, but it should not replace full handwritten reasoning.

7. What is the difference between the two graph lines?

One line plots the original integrand. The other plots the computed antiderivative. Their shapes help you compare how integration changes the function.

8. Can I change the integration constant symbol?

Yes. Enter a custom constant symbol in the form. The displayed result updates the notation while keeping the same mathematical meaning.