Indefinite Integral Calculator

Calculate symbolic indefinite integrals and antiderivatives.

Supports power, logarithmic, trigonometric and exponential integrals, linear substitution, u-substitution patterns and common integration-by-parts problems.

Calculate an Indefinite Integral

Enter an expression using x as the variable.

Examples can use 2x instead of 2*x. Use parentheses for compound expressions such as sin(2x+1).

Try an example:

Supported Input

  • Powers: x^3
  • Negative powers: x^-2
  • Coefficients: 5x^2
  • Addition: x^2+3x
  • Subtraction: x^3-2x
  • Multiplication: x*sin(x)
  • Division: 1/x
  • Parentheses: (x+1)^3
  • Sine: sin(x)
  • Cosine: cos(x)
  • Logarithm: ln(x)
  • Exponential: exp(x)

Supported functions include sin, cos, tan, sec, csc, cot, ln, log, log10, exp and sqrt.

Important Integration Formulas

Power Rule

\[ \int x^n\,dx = \frac{x^{n+1}}{n+1}+C, \qquad n\ne-1 \]

Special Case

\[ \int\frac{1}{x}\,dx = \ln|x|+C \]

Exponential

\[ \int e^x\,dx=e^x+C \]

Sine

\[ \int\sin(x)\,dx=-\cos(x)+C \]

Cosine

\[ \int\cos(x)\,dx=\sin(x)+C \]

Integration by Parts

\[ \int u\,dv = uv-\int v\,du \]

U-Substitution

\[ u=g(x), \qquad du=g'(x)\,dx \]

Indefinite Integral Examples

Polynomial

\[ \int x^3\,dx = \frac{x^4}{4}+C \]

Reciprocal

\[ \int\frac{1}{x}\,dx = \ln|x|+C \]

Trigonometric

\[ \int\sin(x)\,dx = -\cos(x)+C \]

Linear substitution

\[ \int\sin(2x)\,dx = -\frac{1}{2}\cos(2x)+C \]

Exponential substitution

\[ \int e^{3x}\,dx = \frac{1}{3}e^{3x}+C \]

Logarithm

\[ \int\ln(x)\,dx = x\ln(x)-x+C \]

Arctangent pattern

\[ \int\frac{1}{1+x^2}\,dx = \arctan(x)+C \]

Integration by parts

\[ \int x\sin(x)\,dx = -x\cos(x)+\sin(x)+C \]

What Is an Indefinite Integral?

An indefinite integral represents a family of functions whose derivative is the original integrand.

If \[ F'(x)=f(x), \] then: \[ \int f(x)\,dx=F(x)+C. \]

The symbol \(C\) is called the constant of integration. It is necessary because the derivative of every constant is zero.

\[ \frac{d}{dx}[F(x)+C] = F'(x) \]

For this reason, an indefinite integral is also commonly called an antiderivative.

U-Substitution and Reverse Chain Rule

U-substitution is one of the most important methods for evaluating indefinite integrals. It reverses the chain rule from differentiation.

\[ u=g(x) \] \[ du=g'(x)\,dx \]

For example:

\[ \int 2x\cos(x^2)\,dx \] \[ u=x^2 \] \[ du=2x\,dx \] Therefore: \[ \int\cos(u)\,du = \sin(u)+C \] and: \[ \boxed{ \sin(x^2)+C } \]

Integration by Parts

Integration by parts is useful when the integrand is a product of functions and direct substitution is not sufficient.

\[ \int u\,dv = uv-\int v\,du \]

For example:

\[ \int x e^x\,dx \] Choose: \[ u=x, \qquad dv=e^x\,dx \] Then: \[ du=dx, \qquad v=e^x \] Therefore: \[ \int xe^x\,dx = xe^x-\int e^x\,dx \] \[ = xe^x-e^x+C \]

Symbolic vs Numerical Integration

This calculator focuses on symbolic integration. It attempts to return an algebraic antiderivative rather than only a numerical approximation.

For example, symbolic integration gives:

\[ \int e^x\,dx=e^x+C \]

Numerical integration instead calculates an approximate value for a definite integral over specified limits.

For numerical and definite integration, use the related calculators below.

More Integration Calculators

Definite Integral Calculator Integration by Substitution Calculator Integration by Parts Calculator Trigonometric Integral Calculator
Partial Fraction Integral Calculator Numerical Integration Calculator Improper Integral Calculator Graphical Integration Plotter

Related Derivative Calculators

Integration is the inverse operation of differentiation in many important calculus contexts. Continue with these related tools.

Derivative Calculator Higher Order Derivative Calculator Partial Derivative Calculator Derivative at a Point Calculator
Implicit Differentiation Calculator Trigonometric Derivative Calculator Logarithmic & Exponential Derivative Calculator Derivative Graph Plotter

Multivariable Calculus Tools

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Curl Calculator Jacobian Calculator Hessian Calculator

Indefinite Integral Calculator FAQ

What is an indefinite integral?

An indefinite integral represents a family of antiderivatives of a function. The result includes an arbitrary constant C because functions that differ by a constant have the same derivative.

What is an antiderivative?

An antiderivative F(x) of f(x) is a function whose derivative is f(x). In symbols, F'(x)=f(x).

What is the power rule for integration?

For n not equal to -1, the integral of x^n is x^(n+1)/(n+1)+C.

Why is there a +C in an indefinite integral?

The derivative of a constant is zero. Therefore infinitely many functions can have the same derivative, differing only by a constant.

What is the integral of 1/x?

The integral of 1/x is ln|x|+C.

What is the integral of sin(x)?

The integral of sin(x) is -cos(x)+C.

What is the integral of cos(x)?

The integral of cos(x) is sin(x)+C.

What is u-substitution?

U-substitution is a method that replaces a complicated inner expression with a new variable. It is especially useful when the derivative of the inner expression also appears in the integrand.

Can this calculator integrate sin(2x)?

Yes. The calculator recognizes linear substitutions for common trigonometric and exponential functions.

Can this calculator integrate ln(x)?

Yes. The calculator uses integration by parts to obtain x ln(x)-x+C.

Can this calculator solve every integral?

No. Symbolic integration can become very complex. This calculator intentionally supports a defined collection of common elementary integration rules and reports when an expression is outside those rules.

Free Indefinite Integral Calculator

This free indefinite integral calculator is designed to calculate common symbolic antiderivatives. Enter a mathematical expression using \(x\) as the variable and the calculator attempts to find an algebraic function whose derivative is the original expression.

The calculator supports common integration rules including the power rule, constant rule, sum rule, difference rule, constant multiple rule, trigonometric integrals, exponential integrals, logarithmic integrals, linear substitutions and selected integration-by-parts patterns.

Common inputs include x^3, 1/x, sin(x), cos(x), e^x, sin(2x), e^(3x), 1/(1+x^2), and ln(x).

Because symbolic integration can involve very advanced mathematics, the calculator deliberately reports unsupported expressions rather than pretending that an incorrect antiderivative is valid.