0
13.333333
13.333333
2
6.666667
1
0
13.333333
13.333333
2
6.666667
1
The Integral Calculator computes both antiderivatives and definite integrals for polynomials, trigonometric functions, and exponential functions. Integration is one of the two fundamental operations of calculus, alongside differentiation, and it lies at the heart of mathematics, physics, engineering, and economics.
Integration was formalized independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, though the concept of summing infinitely small quantities dates back to Archimedes. The Fundamental Theorem of Calculus establishes the deep connection between integration and differentiation: if $$F(x)$$ is an antiderivative of $$f(x)$$, then the definite integral from $$a$$ to $$b$$ equals $$F(b) - F(a)$$.
This calculator supports four function families. For polynomials of the form $$ax^3 + bx^2 + cx + d$$, the antiderivative is computed using the power rule: $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$$. For sine functions $$a\sin(bx)$$, the antiderivative is $$-\frac{a}{b}\cos(bx) + C$$. For cosine functions $$a\cos(bx)$$, it is $$\frac{a}{b}\sin(bx) + C$$. For exponential functions $$ae^{bx}$$, the antiderivative is $$\frac{a}{b}e^{bx} + C$$.
Applications of integration are vast. In physics, integration computes displacement from velocity, work from force, and electric flux from field strength. In engineering, integrals determine areas, volumes, centroids, and moments of inertia. In probability theory, the integral of a probability density function over an interval gives the probability of the random variable falling in that interval. In economics, the integral of a marginal cost function yields total cost.
Select your function type, enter the coefficients, and specify the bounds to obtain the antiderivative formula, the evaluated antiderivative at both bounds, and the definite integral value. All computations use exact closed-form formulas with no numerical approximation.
The Integral Calculator applies standard antiderivative rules depending on the selected function type.
Polynomial Integration (Power Rule):
$$\int (ax^3 + bx^2 + cx + d)\,dx = \frac{a}{4}x^4 + \frac{b}{3}x^3 + \frac{c}{2}x^2 + dx + C$$
Trigonometric Integration:
$$\int a\sin(bx)\,dx = -\frac{a}{b}\cos(bx) + C$$
$$\int a\cos(bx)\,dx = \frac{a}{b}\sin(bx) + C$$
Exponential Integration:
$$\int ae^{bx}\,dx = \frac{a}{b}e^{bx} + C$$
Definite Integral via the Fundamental Theorem:
$$\int_a^b f(x)\,dx = F(b) - F(a)$$
The calculator evaluates the antiderivative $$F(x)$$ at both the upper and lower bounds, then subtracts to obtain the definite integral. This method is exact for all supported function types.
The Antiderivative F(x) shows the general form of the indefinite integral, including the constant of integration $$C$$. This represents the family of all functions whose derivative equals the original integrand.
The Definite Integral Value is the net signed area between the curve and the x-axis over the interval $$[a, b]$$. Positive values indicate the curve lies predominantly above the x-axis; negative values indicate it lies predominantly below.
F(upper bound) and F(lower bound) show the antiderivative evaluated at each endpoint. Their difference gives the definite integral, providing transparency into the calculation.
Inputs
Results
F(x) = x⁴/4 + 2x³/3 − x²/2 + 3x. F(2) = 16/4 + 16/3 − 4/2 + 6 = 4 + 5.333 − 2 + 6 = 13.333. Wait: 4 + 5.3333 − 2 + 6 = 13.3333. Correction: let's recompute carefully. F(2) = 2⁴/4 + 2·2³/3 − 2²/2 + 3·2 = 4 + 5.3333 − 2 + 6 = 13.3333. F(0) = 0. Definite integral = 13.3333 − 0 = 13.3333.
Inputs
Results
F(x) = (2/3)e^(3x). F(1) = (2/3)e³ = (2/3)(20.0855) = 13.3904. F(0) = (2/3)e⁰ = 2/3 = 0.6667. Definite integral = 13.3904 − 0.6667 = 12.7237.
An integral is the reverse of differentiation. The indefinite integral (antiderivative) of a function $$f(x)$$ is a function $$F(x)$$ such that $$F'(x) = f(x)$$. The definite integral $$\int_a^b f(x)\,dx$$ computes the net signed area between the curve and the x-axis over the interval $$[a, b]$$.
The Fundamental Theorem of Calculus states that if $$F$$ is an antiderivative of a continuous function $$f$$ on $$[a, b]$$, then $$\int_a^b f(x)\,dx = F(b) - F(a)$$. This connects the concept of accumulation (integration) with the concept of rate of change (differentiation).
The constant $$C$$ represents the family of all antiderivatives. Since the derivative of any constant is zero, adding any constant to an antiderivative produces another valid antiderivative. When computing a definite integral, $$C$$ cancels out in the subtraction $$F(b) - F(a)$$.
This calculator handles polynomials up to degree 3, sine and cosine functions of the form $$a\sin(bx)$$ or $$a\cos(bx)$$, and exponentials of the form $$ae^{bx}$$. For more complex functions such as rational functions, logarithmic, or compositions, specialized techniques like substitution or partial fractions are needed.
An indefinite integral produces a function (the antiderivative plus a constant $$C$$), while a definite integral produces a number. The definite integral evaluates the antiderivative at the upper and lower bounds and returns their difference, representing the net signed area under the curve.
A definite integral is negative when the function lies predominantly below the x-axis over the integration interval. The integral measures signed area: portions above the axis contribute positively, and portions below contribute negatively. The result is the net balance of these contributions.
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