Enter a polynomial expression in x and get the antiderivative (indefinite integral) with step-by-step work. Supports polynomial terms, including negative and fractional exponents.
✓ Power rule applied to each term
✓ Negative exponent x^-1 returns ln|x|
✓ Step-by-step breakdown for every term
✓ Fraction coefficients shown exactly, not as decimals
Enter f(x) to IntegrateFree · Instant
∫dx
Format: ax^n terms separated by + or -. Examples: 5x^3 - 2x + 7 or x^-1 + 4x^2
∫ f(x) dx =
F(x) = x^3 + x^2 - 5x + C
Step-by-Step Work
1∫ 3x^2 dx = x^3 [power rule: ∫ x^2 dx = x^3/3]
2∫ 2x dx = x^2 [power rule: ∫ x^1 dx = x^2/2]
3∫ (-5) dx = -5x [constant rule: ∫ a dx = ax]
F(x) = x^3 + x^2 - 5x + C
How Antiderivatives Work
An antiderivative reverses differentiation. If F'(x) = f(x), then F(x) is an antiderivative of f(x). For polynomials, every term follows the power rule for integration.
Power Rule for Integration
∫ ax^n dx = a/(n+1) · x^(n+1) + C (where n ≠ -1)
∫ x^-1 dx = ln|x| + C (special case)
∫ a dx = ax + C (constant rule)
Linearity of Integration
Integration is linear, meaning you can integrate term by term. The antiderivative of a sum is the sum of the antiderivatives. This is why a polynomial expression can be handled one term at a time.
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
Who Is This Calculator For?
Calculus I and II students
Check homework answers and verify power rule application on multi-term polynomials.
Students taking AP Calculus
Practice indefinite integration before AB or BC exams. The step-by-step output shows each term's rule.
Physics and engineering students
Integrate position, velocity, or force functions that appear as polynomial approximations.
Teachers and tutors
Generate worked examples to show the power rule and constant of integration in practice.
How to Use the Calculator
1
Type your expression in the input field
Enter terms in the format ax^n separated by + or -. For example: 4x^3 - 2x^2 + x - 9
2
Use x^n for exponents
Write x^3 for x cubed, x^-1 for 1/x, x^0.5 for the square root. The coefficient goes directly before x with no multiplication sign.
3
Omit the coefficient for 1 or -1
Write x^2 for 1·x^2 and -x^2 for -1·x^2. Including the '1' also works: 1x^2.
4
Read the antiderivative and steps
The result shows F(x) + C and a numbered step for each term, naming the rule applied.
Worked Example: Integrating a Cubic
Find the antiderivative of f(x) = 6x^2 - 4x + 3
Apply power rule to each term:
∫ 6x^2 dx = 6/(2+1) · x^(2+1) = 2x^3
∫ -4x dx = -4/(1+1) · x^(1+1) = -2x^2
∫ 3 dx = 3x (constant rule) = 3x
F(x) = 2x^3 - 2x^2 + 3x + C
Verification: differentiate F(x) = 2x^3 - 2x^2 + 3x + C. The derivative is 6x^2 - 4x + 3, which matches the original f(x). The constant C disappears because d/dx of any constant is zero.
Common Mistakes
!
Adding 1 to the exponent without dividing
The power rule requires both steps: raise the exponent by 1 AND divide the coefficient by the new exponent. Integrating 3x^2 gives x^3, not 3x^3.
!
Using the power rule on x^-1
The power rule breaks down at n = -1 because dividing by n+1 = 0 is undefined. The antiderivative of x^-1 (or 1/x) is ln|x|, not x^0.
!
Forgetting the constant of integration
Every indefinite integral includes an arbitrary constant C. Omitting it means the answer represents only one specific antiderivative rather than the full family.
!
Treating the coefficient as part of the exponent
In 4x^3, the 4 is the coefficient and 3 is the exponent. Applying the power rule gives (4/4)x^4 = x^4, not 4x^4 and not 4/(3+1).
Stewart, J.: Calculus: Early Transcendentals, 9th Edition
Source for power rule, constant of integration, and linearity of indefinite integrals.
2
Larson & Edwards: Calculus, 12th Edition
Reference for integration rules and the special case of ∫ x^-1 dx = ln|x| + C.
D
Dr. Daniel Park
Mathematics lecturer, 11 years teaching calculus at university level
Daniel reviewed the integration logic, edge cases (including x^-1), and worked examples on this page. He teaches single-variable and multivariable calculus.
Reviewed: April 2025
Common Integrals
f(x)
∫ f(x) dx
xⁿ (n ≠ -1)
xⁿ⁺¹/(n+1) + C
x⁻¹ (1/x)
ln|x| + C
a (constant)
ax + C
eˣ
eˣ + C
aˣ
aˣ/ln(a) + C
sin(x)
-cos(x) + C
cos(x)
sin(x) + C
√x
(2/3)x^(3/2) + C
Power Rule Steps
For ax^n (n ≠ -1): 1. Add 1 to the exponent: n → n+1 2. Divide coefficient by the new exponent: a → a/(n+1) 3. Append + C
Example: 5x^3 New exp: 3+1 = 4 New coeff: 5/4 Result: (5/4)x^4 + C
Pro Tip
To verify your antiderivative, differentiate the result. The derivative of F(x) must equal the original f(x). The +C vanishes because the derivative of any constant is zero. This check catches coefficient errors before submitting homework.