0.5
-1
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0
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12
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0.5
-1
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0
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12
4
The Indefinite Integral Calculator computes the antiderivative of a polynomial function $$f(x) = ax^3 + bx^2 + cx + d$$, producing the general antiderivative $$F(x) = \frac{a}{4}x^4 + \frac{b}{3}x^3 + \frac{c}{2}x^2 + dx + C$$. It also evaluates $$F(x)$$ at any point and verifies the result by computing the original function at the same point.
The indefinite integral, also called the antiderivative, reverses the process of differentiation. While differentiation answers "what is the rate of change?" integration answers "what function has this rate of change?" The notation $$\int f(x)\,dx$$ represents the set of all functions whose derivative equals $$f(x)$$. Since the derivative of any constant is zero, antiderivatives are only determined up to an additive constant $$C$$, called the constant of integration.
The power rule for integration is the inverse of the power rule for differentiation. If differentiating $$x^n$$ gives $$nx^{n-1}$$, then integrating $$x^n$$ gives $$\frac{x^{n+1}}{n+1}$$, effectively "undoing" the differentiation. This rule applies to each term of a polynomial independently, thanks to the linearity of integration: $$\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx$$.
Understanding the constant $$C$$ is crucial. In physics, if velocity is given by $$v(t) = 3t^2 + 2t$$, the position function is $$s(t) = t^3 + t^2 + C$$, where $$C$$ is determined by the initial position. Different initial conditions yield different particular solutions from the same general antiderivative. This calculator allows you to specify $$C$$ to explore different members of the antiderivative family.
The verification output computes $$f(x)$$ at the evaluation point, confirming that $$F'(x) = f(x)$$. This provides a built-in consistency check: the derivative of the computed antiderivative should recover the original polynomial. Enter your polynomial coefficients, an evaluation point, and optionally a constant of integration to explore the antiderivative in full detail.
The calculator applies the power rule to each term of $$f(x) = ax^3 + bx^2 + cx + d$$:
$$\int ax^3\,dx = \frac{a}{4}x^4$$
$$\int bx^2\,dx = \frac{b}{3}x^3$$
$$\int cx\,dx = \frac{c}{2}x^2$$
$$\int d\,dx = dx$$
Combining all terms with the constant of integration:
$$F(x) = \frac{a}{4}x^4 + \frac{b}{3}x^3 + \frac{c}{2}x^2 + dx + C$$
The calculator outputs each coefficient of $$F(x)$$ separately (coefficients of $$x^4, x^3, x^2, x$$), evaluates $$F(x)$$ at the specified point, and verifies by computing the original $$f(x)$$ at the same point.
The coefficients of F(x) define the antiderivative polynomial. For instance, if the coefficient of $$x^4$$ is 0.5, the coefficient of $$x^3$$ is −1, and so on, then $$F(x) = 0.5x^4 - x^3 + \ldots + C$$.
F(x) at given x is the antiderivative evaluated at your chosen point, including the constant $$C$$. This is useful for computing particular position, displacement, or accumulated quantities.
The verification value f(x) shows the original function evaluated at the same point. Differentiating $$F(x)$$ should return $$f(x)$$, so this serves as a consistency check on the computation.
Inputs
Results
F(x) = 0.5x⁴ − x³ + 3x² − 4x + C. F(2) = 0.5(16) − 8 + 12 − 8 + 0 = 8 − 8 + 12 − 8 = 4. Verification: f(2) = 2(8) − 3(4) + 6(2) − 4 = 16 − 12 + 12 − 4 = 12. Wait — f(2) = 2·8 − 3·4 + 6·2 − 4 = 16 − 12 + 12 − 4 = 12.
Inputs
Results
F(x) = −(1/3)x³ + 4x + 5. F(3) = −9 + 12 + 5 = 8. Verification: f(3) = −9 + 4 = −5.
An indefinite integral (antiderivative) of $$f(x)$$ is any function $$F(x)$$ such that $$F'(x) = f(x)$$. The general indefinite integral is written as $$\int f(x)\,dx = F(x) + C$$, where $$C$$ is an arbitrary constant.
The constant $$C$$ accounts for the fact that infinitely many functions share the same derivative. In applications, $$C$$ is determined by initial conditions. For example, knowing that position is 10 meters at time zero lets you solve for $$C$$ in the position function.
Differentiate the antiderivative $$F(x)$$ and check that the result equals the original function $$f(x)$$. This calculator does this automatically by computing $$f(x)$$ at the evaluation point, which should be the derivative of $$F(x)$$ at that point.
The power rule states that $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$$ for any real number $$n \neq -1$$. For polynomials, this rule is applied term by term because integration is linear.
Yes, always. Integrating a polynomial of degree $$n$$ produces a polynomial of degree $$n + 1$$. A cubic input gives a quartic antiderivative, a quadratic gives a cubic, and so on. This is a direct consequence of the power rule.
If $$f(x) = 0$$, then every constant function $$F(x) = C$$ is an antiderivative. The calculator will output all zero coefficients and $$F(x) = C$$, which is correct: the integral of zero is a constant.
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