Integral Calculator
Find the antiderivative of a function with the method shown (substitution, by parts, partial fractions), and evaluate definite integrals exactly where possible, including improper integrals to ∞; also the area between two curves and Riemann sums.
About the Integral Calculator
The calculator first tries the standard methods: standard integrals, substitution, integration by parts, partial fractions and trig identities. It names the method at every step, and each step is checked by differentiating the answer back to the integrand.
When none of those methods fits, a symbolic engine finds the antiderivative instead, and its answer is checked the same way. Definite integrals use F(b) − F(a), compared with a numerical integration so a jump inside the interval can't give a wrong answer.
The formula
- By parts
- ∫ u dv = uv − ∫ v du
- Substitution
- ∫ f(g(x)) g′(x) dx = ∫ f(u) du
Standard integrals
| f(x) | ∫ f(x) dx |
|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n + 1) |
| 1/x | ln|x| |
| eˣ | eˣ |
| sin x | −cos x |
| cos x | sin x |
| sec² x | tan x |
| 1/(1 + x²) | tan⁻¹ x |
| 1/√(1 − x²) | sin⁻¹ x |
Worked example
The area under f(x) = 2x + 3 from x = 0 to x = 2
- Antiderivative: x² + 3x
- At 2: 4 + 6 = 10; at 0: 0
- Area = 10 − 0 = 10
Questions people ask
Why add + C?
Any constant differentiates to 0, so every antiderivative can be shifted by a constant.
Why does my textbook's answer look different?
Antiderivatives can differ by a constant or be written differently (for example ln|sec x| = −ln|cos x|). Differentiate both to check.
Can every function be integrated?
Not in elementary terms: e^(−x²) has no such antiderivative. You can still get its definite integral numerically by giving limits.