Online Integral & Antiderivative Calculator with Steps (Definite & Indefinite)

How to Use the Integral Calculator

Solving integrals is simple with our interactive mathematical engine. By using an online antiderivative calculator, you eliminate the guesswork from complex calculus homework:

1. Enter your function: Use the extended math keyboard above or type your expression using standard mathematical notation (e.g., x^2 * sin(x)).
2. Select the Type: Choose Indefinite Integral to find the general antiderivative, or toggle the settings to Definite Integral to explicitly define your upper and lower bounds.
3. Calculate: Click "Solve." The calculator will output the exact symbolic result and provide a detailed calculation results drop-down showing the precise integration rules applied at each step.

Types of Integrals Supported

Definite Integrals

A definite integral calculates the exact numerical signed area under a curve between two specific points on the $x$-axis over a given interval. Originating from the concept of a Riemann Sum, it is represented as: $$\int_{a}^{b} f(x) \, dx$$ You can solve a definite integral online easily. Our calculator solves this by first finding the antiderivative and then applying the Fundamental Theorem of Calculus to compute the numerical value $F(b) - F(a)$, where $a$ is the lower bound and $b$ is the upper bound.

Indefinite Integrals (Antiderivatives)

An indefinite integral represents an entire family of curves whose derivative is the original mathematical function. Because the derivative of a constant is zero, finding the indefinite integral always results in a general solution that includes an arbitrary constant of integration, $+ C$: $$\int f(x) \, dx = F(x) + C$$

Pro Tip: You can always verify your indefinite integral by differentiating the result using our Derivative Calculator.

Improper Integrals

Improper integrals involve calculating the area over an unbounded interval (where one or both limits of integration approach $\infty$ or $-\infty$), or integrating functions across a mathematical singularity or vertical asymptote. The engine automatically handles these limits to determine if the integral converges to a finite value or diverges to infinity.

Common Integration Methods

Our engine doesn't just give you the final answer; it shows you exactly how to solve integrals using standard analytical methods and techniques.

Integration by Parts

Derived from the product rule of differentiation, integration by parts transforms the integral of a product of functions into a simpler integral. It follows the formula: $$\int u \, dv = uv - \int v \, du$$ The calculator explicitly shows the pedagogical process: how to choose $u$ and $dv$ (often applying the LIATE rule), compute the differential $du$, and integrate to find $v$.

U-Substitution

Integration by substitution (or $u$-substitution) is the inverse of the chain rule. It simplifies integrals by performing a change of variables to a new variable, $u$, where $du$ represents the derivative of the inner function.

Partial Fractions

When integrating complex rational functions (a polynomial divided by a polynomial), the calculator performs a partial fraction decomposition. It breaks the denominator into roots, linear factors, and irreducible quadratic factors, based on the polynomial degree, creating simpler fractions that are easy to integrate.

Real-World Applications of Integrals

Calculus is not just an abstract mathematical concept; integrals have major applications in the physical world. Understanding these applications is important for students of physics, engineering, and economics.

• Kinematics: If you are given a function representing the velocity of an object over time, taking the definite integral of that function will yield the total displacement (distance traveled) of the object.
• Physics (Work and Force): In mechanics, if a variable force is applied to an object over a certain distance, the total work done is calculated by taking the integral of the force function with respect to distance ($W = \int F(x) dx$).
• Economics: Integrals are frequently used in economics to calculate Consumer Surplus and Producer Surplus, by determining the area between the supply/demand curves and the equilibrium price.

Step-by-Step Worked Examples

Practicing with worked examples is the best way to learn mathematical proofs and calculus techniques. Here are detailed breakdowns of common problems:

Example 1: Indefinite Integral Using Integration by Parts
Solve $\int x \cos(x) \, dx$

1. Identify the method: Since this is a product of an algebraic function ($x$) and a trigonometric function ($\cos(x)$), we use integration by parts.
2. Apply LIATE rule: Let $u = x$ (Algebraic) and $dv = \cos(x) \, dx$ (Trig).
3. Differentiate and Integrate: Then $du = 1 \, dx$ and $v = \sin(x)$.
4. Substitute into formula: Using $\int u \, dv = uv - \int v \, du$, we get $x \sin(x) - \int \sin(x) \, dx$.
5. Final Answer: $x \sin(x) + \cos(x) + C$.

Example 2: U-Substitution
Solve $\int 2x e^{x^2} \, dx$

1. Identify the inner function: Let $u = x^2$.
2. Find the derivative: Then $du = 2x \, dx$. Notice that $2x \, dx$ perfectly matches the rest of our integral.
3. Substitute: The integral simplifies entirely to $\int e^u \, du$.
4. Integrate: The integral of $e^u$ is simply $e^u$.
5. Final Answer: Substitute $x^2$ back in to get $e^{x^2} + C$.

Example 3: Evaluating a Definite Integral
Solve $\int_{0}^{2} 3x^2 \, dx$

1. Find the Antiderivative: Applying the power rule of integration, the antiderivative of $3x^2$ is $x^3$.
2. Apply the Bounds: We evaluate $x^3$ from the lower bound $0$ to the upper bound $2$.
3. Calculate: $(2)^3 - (0)^3 = 8 - 0 = 8$.

Supported Functions and Constants (Syntax Guide)

Our calculator's parser supports all standard mathematical notation. You can use the clickable UI keyboard or type these directly:

• Trigonometric: sin(x), cos(x), tan(x), sec(x), csc(x), cot(x)
• Inverse Trig: arcsin(x), arccos(x), arctan(x)
• Hyperbolic: sinh(x), cosh(x), tanh(x)
• Logarithmic & Exponential: ln(x) (natural log), log(x) (base 10), log2(x), exp(x) or e^x
• Constants & Symbols: pi ($\pi$), e, abs(x) (absolute value), sqrt(x) (square root)