Antiderivative of Trig Functions: Sin, Cos, Tan, Sec, Csc, Cot
Antiderivatives of all six trig functions with proofs, worked examples, and common mistakes.
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Antiderivative Calculator 2026
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How to Calculate Antiderivative
An antiderivative reverses differentiation. If F'(x) = f(x), then F(x) is an antiderivative of f(x). For any polynomial, every term follows three integration rules:
Power rule: ∫ ax^n dx = a/(n+1) x^(n+1) + C (n not equal to -1)
Log rule: ∫ x^-1 dx = ln|x| + C (special case for n = -1)
Constant rule: ∫ a dx = ax + C (a is any number)
Integration is linear, meaning you can integrate term by term. The antiderivative of a sum equals the sum of the antiderivatives:
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
This linearity is why a polynomial with five terms produces five separate integration steps. For polynomial function analysis using a different approach, the Interpolation Calculator fits a polynomial through data points, which connects naturally to integral approximation.
Antiderivative Calculator with Steps: How Each Rule Works
For every term in the expression, the calculator identifies the rule, applies it, and labels the step. The table below shows the pattern for the most common term types:
| f(x) term | Rule applied | F(x) result | Verification: F'(x) |
|---|---|---|---|
| 6x^2 | Power: exp 2+1=3, coeff 6/3=2 | 2x^3 | 6x^2 ✓ |
| -4x | Power: exp 1+1=2, coeff -4/2=-2 | -2x^2 | -4x ✓ |
| 3 | Constant: ∫ a dx = ax | 3x | 3 ✓ |
| 5x^-2 | Power: exp -2+1=-1, coeff 5/-1=-5 | -5x^-1 | 5x^-2 ✓ |
| x^-1 | Log rule (special case) | ln|x| | 1/x ✓ |
| x^0.5 | Power: exp 0.5+1=1.5, coeff 1/1.5=2/3 | (2/3)x^1.5 | x^0.5 ✓ |
The verification column confirms each result by differentiating. This is the fastest way to check antiderivative work: differentiate F(x), and the result must match f(x) exactly. For coordinate geometry calculations that also involve verifying results step by step, the Midpoint Calculator covers the midpoint formula with full working shown.
The Most General Antiderivative and the Constant of Integration
The most general antiderivative of f(x) is F(x) + C, where C is an arbitrary constant. This single expression represents infinitely many functions, all with the same derivative. For example, the most general antiderivative of 2x includes x^2, x^2 + 7, x^2 - 100, and every other vertical shift.
Most general antiderivative of 3x^2: x^3 + C
Includes: x^3, x^3 + 1, x^3 - 4, x^3 + 1000, ...
All have derivative: 3x^2
Includes: x^3, x^3 + 1, x^3 - 4, x^3 + 1000, ...
All have derivative: 3x^2
Particular antiderivative: When an initial condition is given (such as F(2) = 10), C is determined and the result is a specific function rather than a family. This is the basis of initial value problems in differential equations.
Find F(x) if F'(x) = 2x and F(0) = 5
General: F(x) = x^2 + C
Apply condition: F(0) = 0^2 + C = 5, so C = 5
Particular: F(x) = x^2 + 5
General: F(x) = x^2 + C
Apply condition: F(0) = 0^2 + C = 5, so C = 5
Particular: F(x) = x^2 + 5
Second antiderivatives follow the same pattern but introduce a second constant. If acceleration a(t) = 6t, velocity is v(t) = 3t^2 + C1, and position is s(t) = t^3 + C1t + C2. Each integration step adds one new constant, determined by an additional initial condition. For data analysis applications where multiple constants are fitted to observations, the Correlation Coefficient Calculator shows how constants are determined from data points.
Worked Example: Integrating a Cubic Polynomial
Find the most general 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 f(x) exactly. The constant C disappears because the derivative of any constant is zero.
Antiderivative Mistakes That Produce Wrong Coefficients, Exponents, or Missing Constants
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.
Applying the power rule to x^-1
The power rule breaks down at n = -1 because dividing by n+1 = 0 is undefined. The antiderivative of x^-1 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 general family.
Confusing the coefficient with the exponent
In 4x^3, the 4 is the coefficient and 3 is the exponent. The power rule gives (4/4)x^4 = x^4, not 4x^4 or 4/(3+1)x^4.
Differentiating instead of integrating
The derivative of x^3 is 3x^2 (multiply by exponent, reduce exponent by 1). The antiderivative is x^4/4 (add 1 to exponent, divide by new exponent). Mixing these operations is the most common error when switching between differentiation and integration.
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Sources & References
1
OpenStax Calculus Volume 1 - Antiderivatives ↗
Free peer-reviewed calculus textbook covering the power rule for integration, the constant of integration, and the linearity of indefinite integrals.
2
Khan Academy - Indefinite Integrals Review ↗
Reference for integration rules including the power rule, constant rule, and the special case of the antiderivative of x^-1 = ln|x| + C.
3
MIT OpenCourseWare: 18.01 Single Variable Calculus ↗
Open lecture notes covering indefinite integrals, the power rule, antiderivative techniques, and initial value problems.
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Hassaan Rasheed
Developer and Researcher, CalculatorFlux
Researches and verifies the formulas, methodology, and source data behind each calculator on CalculatorFlux. All tools are built and checked against the cited references before publication.
Last updated: May 2026