Advanced Math

Integral Calculator

Integrates sums and constant multiples of powers of x (including 1/x and sqrt(x)) and sin, cos and e^x of a linear expression. Indefinite answers include + C; definite integrals show F(b) − F(a) with the bounds substituted. Anything needing substitution or integration by parts is reported as unsupported rather than guessed.

Inputs∫ f(x) dx

Typef(x)Examples: 4x^3 − x + 2 · (x + 1)^2 · 1/x · sqrt(x) · sin(2x) · 3cos(x) · e^(3x). Not supported: products like x·sin(x) or insides like sin(x^2).

Step-by-step

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Fill in the fields and press Solve. The full method will appear here, one numbered step at a time.

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Worked example

Evaluate the definite integral of 3x² + 2x from 0 to 2.

1.
Start with the integral.
```
∫ (3x^2 + 2x) dx
```
2.
Sum rule: integrate each term separately.
3.
Power rule: ∫x^n dx = x^(n+1)/(n+1), here n = 2, so n + 1 = 3. Constant multiples stay in front.
```
∫ 3x^2 dx = x^3
```
4.
Power rule: ∫x^n dx = x^(n+1)/(n+1), here n = 1, so n + 1 = 2. Constant multiples stay in front.
```
∫ 2x dx = x^2
```
5.
Write the antiderivative F(x). (The + C cancels in a definite integral.)
```
F(x) = x^3 + x^2
```
6.
Fundamental theorem of calculus: evaluate F at the upper bound minus F at the lower bound.
```
∫ from 0 to 2 = F(2) − F(0)
```
7.
Substitute x = 2.
```
F(2) = 12
```
8.
Substitute x = 0.
```
F(0) = 0
```
9.
Subtract.
```
12 − (0) = 12
```

Answer: 12

Note: Powers of x, polynomials, and sin, cos or e of a linear inside. No integration by parts or substitution.

Use “Load worked example” above to run this in the solver.

Related formulas

The formulas this tool uses, each explained with symbols, conditions and its own example in the Formula Library.

Power rule (integrals)∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C Integral of 1/x∫ 1/x dx = ln|x| + C Definite integral∫ₐᵇ f(x) dx = F(b) − F(a)