Enter Integrand
Result
Indefinite result includes an arbitrary constant C. If bounds are provided, the definite value uses the Fundamental Theorem of Calculus.
Antiderivative \( \int f(x)\,dx \):
Formulas Used
- Linearity \( \int(\alpha f+\beta g)\,dx=\alpha\int f\,dx+\beta\int g\,dx \)
- Power rule \( \int x^n dx=\frac{x^{n+1}}{n+1}+C\) for \(n\\neq-1\)
- Reciprocal \( \int \\frac{1}{x}\,dx=\ln|x|+C \)
- Linear denominator \( \int \\frac{1}{ax+b}\,dx=\\frac{1}{a}\ln|ax+b|+C \)
- Exponential \( \int e^{ax}\,dx=\\frac{1}{a}e^{ax}+C \)
- Trigonometric
\( \int \\sin(ax)\,dx=-\\frac{1}{a}\\cos(ax)+C \), \( \int \\cos(ax)\,dx=\\frac{1}{a}\\sin(ax)+C \)
\( \int \\sec^2(ax)\,dx=\\frac{1}{a}\\tan(ax)+C \), \( \int \\csc^2(ax)\,dx=-\\frac{1}{a}\\cot(ax)+C \)
\( \int \\sec(ax)\\tan(ax)\,dx=\\frac{1}{a}\\sec(ax)+C \), \( \int \\csc(ax)\\cot(ax)\,dx=-\\frac{1}{a}\\csc(ax)+C \) - Chain factors For \(u=ax+b\), substitute to account for constant \(a\).
The engine combines rule-based recognition with a symbolic computer algebra system for broader coverage.
How to Use
- Enter the integrand using explicit multiplication, e.g.,
3*x^2. - Use
e^(...),sin(...),cos(...),tan(...),log(...)for natural log. - Optionally supply bounds
aandbto compute a definite value. - Press Compute to see the antiderivative and reasoning steps.
- Click Add to History then export CSV or PDF for records.
Tip: Implicit products like
3x should be written 3*x. Use log for \( \ln \).
Example Data
| # | Integrand | Variable | Lower | Upper |
|---|
Quick Reference — Common Antiderivatives
| # | \(f(x)\) | \(F(x)+C\) | Notes |
|---|---|---|---|
| 1 | \(x^n,\, n\\neq-1\) | \(\\dfrac{x^{n+1}}{n+1}+C\) | Power rule |
| 2 | \(\\dfrac{1}{x}\) | \(\\ln|x|+C\) | Power rule exception |
| 3 | \(e^{ax}\) | \(\\dfrac{1}{a}e^{ax}+C\) | \(a\\neq0\) |
| 4 | \(\\sin(ax)\) | \(-\\dfrac{1}{a}\\cos(ax)+C\) | \(a\\neq0\) |
| 5 | \(\\cos(ax)\) | \(\\dfrac{1}{a}\\sin(ax)+C\) | \(a\\neq0\) |
| 6 | \(\\sec^2(ax)\) | \(\\dfrac{1}{a}\\tan(ax)+C\) | Derivative pair |
| 7 | \(\\csc^2(ax)\) | \(-\\dfrac{1}{a}\\cot(ax)+C\) | Derivative pair |
| 8 | \(\\sec(ax)\\tan(ax)\) | \(\\dfrac{1}{a}\\sec(ax)+C\) | Derivative pair |
| 9 | \(\\csc(ax)\\cot(ax)\) | \(-\\dfrac{1}{a}\\csc(ax)+C\) | Derivative pair |
| 10 | \(\\dfrac{1}{ax+b}\) | \(\\dfrac{1}{a}\\ln|ax+b|+C\) | Linear denominator |
| 11 | \(k\\cdot f(x)\) | \(k\\int f(x)dx\) | Linearity |
| 12 | \(f(x)+g(x)\) | \(\\int f+\\int g\) | Linearity |
Benchmark Test Cases — Expected Antiderivatives
| # | Input \(f(x)\) | Expected \(F(x)+C\) | Comments |
|---|---|---|---|
| 1 | \(3x^2\) | \(x^3+C\) | Basic power rule |
| 2 | \(2e^{3x}\) | \(\\dfrac{2}{3}e^{3x}+C\) | Exponential with scale |
| 3 | \(\\sin(2x)-\\dfrac{4}{x}\) | \(-\\dfrac{1}{2}\\cos(2x)-4\\ln|x|+C\) | Sum of standard forms |
| 4 | \(\\dfrac{1}{2x+5}\) | \(\\dfrac{1}{2}\\ln|2x+5|+C\) | Linear denominator |
| 5 | \(\\sec^2(4x)\) | \(\\dfrac{1}{4}\\tan(4x)+C\) | Derivative pair |
| 6 | \(7\) | \(7x+C\) | Constant rule |
| 7 | \(\\cos(5x)+3x^2\) | \(\\dfrac{1}{5}\\sin(5x)+x^3+C\) | Linearity check |
| 8 | \(\\tan x\) | \(-\\ln|\\cos x|+C\) | Identity-based |
Use these cases to verify correctness. The app also differentiates \(F\) to compare with \(f\).
History
| # | Timestamp | Integrand | Var | Antiderivative F(x)+C | Definite value |
|---|
FAQs
It means finding an antiderivative \(F\) such that \(F' = f\). This is the inverse operation of differentiation and is also called an indefinite integral.
Polynomials, exponentials, logarithms, trigonometric functions, sums, constant multiples, simple rational forms like \(1/x\) and \(1/(ax+b)\), and many compositions. The CAS backend broadens coverage further.
The parser and CAS expect explicit products like
3*x. Implicit forms such as 3x are ambiguous for reliable symbolic processing.
Yes. Provide bounds \(a\) and \(b\). The tool first finds an antiderivative \(F\), then evaluates \(F(b)-F(a)\) using symbolic substitution and numeric evaluation for a final value.
Steps are generated by pattern recognition when possible, otherwise summarized by CAS. For complex forms, steps may be schematic rather than textbook-perfect derivations.
The calculator differentiates the found \(F\) and simplifies to compare with the input \(f\). A success badge appears when the derivative simplifies to the original integrand.
Use
log for the natural logarithm, write products explicitly, group with parentheses, and avoid absolute-value bars; the system writes them where necessary for logs of linear terms.