Calculator
Example Data Table
| Integrand f(x) | One antiderivative F(x) | Rule used |
|---|---|---|
| 3*x^2 + 4 | x^3 + 4*x + C | Power + constant |
| sin(2*x) | -cos(2*x)/2 + C | Trig with linear inner |
| exp(3*x-1) | exp(3*x-1)/3 + C | Exponential with linear inner |
| 1/x | ln(abs(x)) + C | Log rule |
| ln(5*x+2) | ((5*x+2)*ln(5*x+2)-(5*x+2))/5 + C | Parts template (linear) |
Formula Used
- Linearity: ∫(f±g) dx = ∫f dx ± ∫g dx, and ∫k·f dx = k·∫f dx.
- Power rule: ∫xp dx = xp+1/(p+1), for p ≠ −1.
- Log rule: ∫1/x dx = ln|x|.
- Basic trig: ∫sin(x) dx = −cos(x), and ∫cos(x) dx = sin(x).
- Exponential: ∫eu dx = eu/u′ when u is linear in x.
- Linear inner (chain): if u = ax + b, then du/dx = a, so dividing by a adjusts the antiderivative.
How to Use This Calculator
- Type your function using x (or pick another variable).
- Keep multiplication explicit, like 3*x not 3x.
- Use ^ for powers, for example x^3.
- For functions, write sin(2*x), exp(x), ln(x).
- Click Compute Integral to see the antiderivative and checks.
- Use downloads to save the solution as CSV or PDF.
Article
1) Purpose of an Indefinite Integral Tool
This calculator turns a typed function f(x) into an antiderivative F(x). It aims at fast practice, clean notation, and quick checking for common calculus coursework. Instead of returning only a number, it returns a new function plus an optional constant C.
2) Expression Entry and Accepted Notation
Use explicit multiplication like 2*x and powers like x^3. Write functions as sin(x), cos(x), tan(x), exp(x), ln(x) or log(x). Constants pi and e are allowed. Keeping parentheses clear reduces parsing mistakes and improves matches.
3) Rules Applied to Produce an Antiderivative
The engine uses linearity to split sums and pull out constants. It applies the power rule for x^p when p is not −1, and the logarithm rule for 1/x. It also includes basic trig and exponential patterns, especially when the inside is linear. These templates cover many exam questions, including mixed polynomials with trig and exponentials.
4) Linear Inner Forms and Chain Adjustments
When the inside looks like a*x+b, the derivative is the constant a. So the tool divides the usual antiderivative by a to compensate. That is why sin(2*x) becomes −cos(2*x)/2 and exp(3*x−1) becomes exp(3*x−1)/3. The same idea supports cos(5*x) and ln(7*x+4) style inputs too.
5) Step Text and Simplified Output
If steps are enabled, you see which rule triggered for each term. Simplification removes extra zeros and ones, and it formats nested expressions consistently. You can also toggle the + C option to match typical textbook answers. This helps you compare your manual work line by line without rewriting the whole solution.
6) Numeric Derivative Verification
After computing F(x), the calculator estimates F′(x) using a small finite difference. It compares that estimate with f(x) at several sample points and reports the error. Tiny errors are expected from rounding, but large errors suggest an unsupported form. Think of it as a quick sanity check, not a formal proof, and use it to catch typos early.
7) Plotting and Downloads
Plotting draws both f(x) and one representative F(x) across your chosen range. Because different antiderivatives differ by a constant, the F(x) curve may appear shifted vertically. Download buttons export your input, variable, result, and optional steps to CSV or PDF for reuse. Use CSV for sheets and PDF for printouts.
FAQs
1) Why is “+ C” optional here?
Indefinite integrals represent a family of antiderivatives. Adding any constant does not change the derivative. Turn it on for textbook style, or off when you only need one representative function.
2) Which inputs work best?
Polynomials, 1/x, sin, cos, tan, exp, and ln/log work best. Linear inner forms like sin(2*x), exp(3*x-1), and ln(5*x+2) are supported reliably.
3) What does “simplify output” change?
It removes obvious extras like multiplying by 1, adding 0, and repeated parentheses. The math stays equivalent, but the final antiderivative becomes easier to read and compare.
4) Why can the derivative check show nonzero error?
The check uses finite differences, so rounding and step size introduce small numeric error. If the error is consistently large, re-check your syntax or try rewriting the expression into simpler terms.
5) Why does F(x) look vertically shifted on the plot?
Any antiderivative can be shifted by a constant C. The plot shows one representative antiderivative with C omitted, so vertical shifts are normal and do not indicate a mistake.
6) How do the downloads help?
CSV is handy for spreadsheets, logs, and batch reviewing. PDF is better for printing, sharing, and attaching to assignments. Both exports include your input, variable, result, and optional steps.
Article
1) Purpose of an Indefinite Integral Tool
This calculator turns a typed function f(x) into an antiderivative F(x). It aims at fast practice, clean notation, and quick checking for common calculus coursework. Instead of returning only a number, it returns a new function plus an optional constant C.
2) Expression Entry and Accepted Notation
Use explicit multiplication like 2*x and powers like x^3. Write functions as sin(x), cos(x), tan(x), exp(x), ln(x) or log(x). Constants pi and e are allowed. Keeping parentheses clear reduces parsing mistakes and improves matches.
3) Rules Applied to Produce an Antiderivative
The engine uses linearity to split sums and pull out constants. It applies the power rule for x^p when p is not −1, and the logarithm rule for 1/x. It also includes basic trig and exponential patterns, especially when the inside is linear. These templates cover many exam questions, including mixed polynomials with trig and exponentials.
4) Linear Inner Forms and Chain Adjustments
When the inside looks like a*x+b, the derivative is the constant a. So the tool divides the usual antiderivative by a to compensate. That is why sin(2*x) becomes −cos(2*x)/2 and exp(3*x−1) becomes exp(3*x−1)/3. The same idea supports cos(5*x) and ln(7*x+4) style inputs too.
5) Step Text and Simplified Output
If steps are enabled, you see which rule triggered for each term. Simplification removes extra zeros and ones, and it formats nested expressions consistently. You can also toggle the + C option to match typical textbook answers. This helps you compare your manual work line by line without rewriting the whole solution.
6) Numeric Derivative Verification
After computing F(x), the calculator estimates F′(x) using a small finite difference. It compares that estimate with f(x) at several sample points and reports the error. Tiny errors are expected from rounding, but large errors suggest an unsupported form. Think of it as a quick sanity check, not a formal proof, and use it to catch typos early.
7) Plotting and Downloads
Plotting draws both f(x) and one representative F(x) across your chosen range. Because different antiderivatives differ by a constant, the F(x) curve may appear shifted vertically. Download buttons export your input, variable, result, and optional steps to CSV or PDF for reuse. Use CSV for sheets and PDF for printouts.
FAQs
1) Why is “+ C” optional here?
Indefinite integrals represent a family of antiderivatives. Adding any constant does not change the derivative. Turn it on for textbook style, or off when you only need one representative function.
2) Which inputs work best?
Polynomials, 1/x, sin, cos, tan, exp, and ln/log work best. Linear inner forms like sin(2*x), exp(3*x-1), and ln(5*x+2) are supported reliably.
3) What does “simplify output” change?
It removes obvious extras like multiplying by 1, adding 0, and repeated parentheses. The math stays equivalent, but the final antiderivative becomes easier to read and compare.
4) Why can the derivative check show nonzero error?
The check uses finite differences, so rounding and step size introduce small numeric error. If the error is consistently large, re-check your syntax or try rewriting the expression into simpler terms.
5) Why does F(x) look vertically shifted on the plot?
Any antiderivative can be shifted by a constant C. The plot shows one representative antiderivative with C omitted, so vertical shifts are normal and do not indicate a mistake.
6) How do the downloads help?
CSV is handy for spreadsheets, logs, and batch reviewing. PDF is better for printing, sharing, and attaching to assignments. Both exports include your input, variable, result, and optional steps.