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Basic Integration Formulas Online Calculator

Basic integral formula lookup and verifier. Built-in integrals for powers, 1/x, e^x, trig and inverse trig—pick one, plug in values and verify numerically to learn the table and check homework.

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∫ x^n dx = x^(n+1)/(n+1) + C
Choose a preset to fill in f(x) and interval.
Supports + - * / ^ ( ) and sin cos tan sec cot sqrt exp log abs asin atan pi e; write 2*x, not 2x
Lower limit of integration
Upper limit of integration

📖 Tutorial | Basic Integration Formulas

1. Definition
Basic integration formulas are reversed from the basic derivative formulas and are the building blocks of all integration. Complex integrals are reduced to these forms by substitution and by parts.
2. Standard table
SymbolsMeaning
∫xⁿ dxx^(n+1)/(n+1)+C (n≠−1)
∫1/x dxln|x|+C
∫eˣ dxeˣ+C
∫sin x dx−cos x+C
∫cos x dxsin x+C
∫sec²x dxtan x+C
∫1/(1+x²)dxatan x+C
3. How it works
  • Pick a standard form from the dropdown; f(x) and bounds are filled in;
  • Adaptive Simpson integrates ∫_a^b f(x)dx numerically;
  • Compare the numeric result with the analytic formula.
4. Steps
  1. Choose a standard integral from the dropdown;
  2. Fields fill automatically; edit them if you wish;
  3. Click Calculate to see the numeric vs. analytic result.
5. Example
Example: ∫₀^π sin x dx = [−cos x]₀^π = 2. The tool returns 2.000000.
6. Pitfalls
∫1/x dx is ln|x|; do not drop the absolute value;
∫xⁿ needs n≠−1; n=−1 is the special ln|x| case;
Numeric results are for verification; use analytic formulas in formal work.

❓ FAQ | Basic Integration Formulas

Why does ∫xⁿ require n≠−1?
At n=−1 the result is ln|x| and the power formula divides by zero; it must be memorized separately.
How to memorize them?
Mirror the derivative rules: integration raises the power by one and divides.
Can I edit after choosing a preset?
Yes. Presets fill the fields; you can still edit f(x) and the bounds.
Why is the numeric result not exact?
Adaptive integration has about 1e-6 error, which is normal.
What notation is supported?
sin/cos/exp/log/sec/sqrt/pi/e; never omit the multiplication sign.