This online calculator will try to find the indefinite integral (antiderivative) of the given function, with steps shown. Different techniques are used: integration by substitution, integration by parts, integration by partial fractions, trigonometric substitutions, etc.
Related calculator: Definite and Improper Integral Calculator
The Integral Calculator is an indispensable online assistant for working with indefinite integrals. Our user-friendly digital platform lets you calculate integrals and also provides step-by-step solutions to improve your skills in taking integrals.
How to Use the Integral Calculator?
Input
Begin by entering the function you want to integrate. You can input a wide range of functions, from simple to complex.
Calculation
Once you've entered the function and, if necessary, the variable, click the "Calculate" button.
Result
The calculator will immediately display (if it can calculate) the value of the integral.
What Is an Integral?
The integral is a fundamental concept in calculus that allows us to solve various mathematical and real-world problems. There are two main types of integrals: the indefinite integral and the definite integral.
Indefinite Integral (Antiderivative)
The indefinite integral, often called the antiderivative, doesn't have specific bounds. It is an operation of finding such a function $$$F(x)$$$ whose derivative equals a given function $$$f(x)$$$. Mathematically, the indefinite integral can be written as follows:
$$\int f(x)dx=F(x)+C,$$
where:
- $$$f(x)$$$ is the integrand, i.e. the function to be integrated.
- $$$F(x)$$$ is the antiderivative of $$$f(x)$$$.
- $$$C$$$ is the constant of integration, which accounts for all possible antiderivatives: if $$$F^{\prime}(x)=f(x)$$$, then $$$\left(F(x)+C\right)^{\prime}=f(x)$$$.
For example, let's find the antiderivative of $$$f(x)=2x$$$. This can be written as follows:
$$\int 2xdx$$
There are many techniques for taking integrals, but this integral requires the constant multiple rule and the power rule:
$$\int 2xdx=2\int xdx=2\int x^1dx=2\frac{x^{1+1}}{1+1}=x^2+C$$
So the antiderivative of $$$2x$$$ is $$$x^2+C$$$, where $$$C$$$ represents the constant of integration.
Definite Integral
The definite integral, denoted by $$$\int_a^b f(x)dx$$$, represents the accumulated quantity or the net area between the curve of the function $$$f(x)$$$ and the x-axis over the interval $$$[a, b]$$$. According to the Fundamental Theorem of Calculus, there is a connection between the definite integral and the antiderivative:
$$\int_a^b f(x)dx=F(b)-F(a),$$
where:
- $$$a$$$ and $$$b$$$ are the lower and upper bounds of the integration.
- $$$f(x)$$$ is the integrand, representing the function to be integrated.
- $$$F(x)$$$ is the antiderivative (indefinite integral) of $$$f(x)$$$.
For example, let's find the area below the curve $$$f(x)=x^2$$$ and above the x-axis between $$$x=1$$$ and $$$x=3$$$. We can use the definite integral:
$$\int_1^3x^2dx$$
To evaluate this, we first find the antiderivative of $$$x^2$$$, which is $$$\frac{x^{2+1}}{2+1}=\frac{x^3}{3}$$$. Now, we apply the Fundamental Theorem of Calculus:
$$\int_1^3x^2dx=\left.\frac{x^3}{3}\right|_1^3=\frac{3^3}{3}-\frac{1^3}{3}=\frac{26}{3}$$
So the area under the curve $$$f(x)=x^2$$$ between $$$x=1$$$ and $$$x=3$$$ is $$$26$$$ square units.
The integral is an important tool to solve various mathematical problems. The indefinite integral helps us understand the relationship between the function and its antiderivative, and the definite integral allows us to calculate areas, accumulate quantities, etc. over certain intervals.
Why Choose Our Integral Calculator?
Accuracy
Our Integral Calculator is designed to provide accurate results. It removes the risk of calculation errors, ensuring that you always obtain the correct integral value.
Versatility
The calculator can handle a wide range of functions, from simple to complex.
Step-by-Step Solutions
By providing step-by-step solutions, the tool can help deepen your knowledge of integrals.
User-Friendly Interface
The platform is intuitive and user-friendly. You don't need to be a calculus expert to use it. Simply input your function, and the calculator will do the rest.
Accessibility
The Integral Calculator is available to you at no cost. You can use it whenever you need it without any subscription or payment requirements.
FAQ
What are the common integration techniques?
The common integration techniques include:
- integration by substitution
- integration by parts
- integration by partial fractions
- trigonometric substitutions
Can you take numbers out of an integral?
Yes, you can take constants or numbers out of an integral. This property is known as the constant multiple rule of integration. When you have an integral of the form $$$\int cf(x)dx$$$, where $$$c$$$ is a constant or any expression that does not depend on $$$x$$$, you can factor $$$c$$$ out of the integral: $$$\int cf(x)dx=c\int f(x)dx$$$. It allows you to simplify the integration process when dealing with constants.
Can an integral be infinite?
Yes, a definite integral can be infinite under certain conditions. For example, if you have an improper integral with unbounded integration limits, or if the integrated function has an infinite discontinuity within an interval of integration, the integral may diverge, i.e. it may not have a finite value. For example, $$$\int_{-1}^1\frac{dx}{x}$$$ diverges.
Can an integral be zero?
Yes, an integral can be evaluated to zero. When the area under a curve, as calculated by the integral, is equal to zero, it means that the positive and negative areas cancel each other out within the given interval. For example, $$$\int_{-2}^2x^3dx=0$$$.
What is a double integral?
The double integral is an extension of the concept of integration to two-dimensional space. It involves integrating a two-variable function over a specified region in the xy-plane. Double integrals are used to calculate areas, volumes, and other quantities in two-dimensional space. The notation for a double integral is $$$\iint f(x,y)dA$$$.
What is an integral value?
There are two types of integral values: the indefinite integral value and the definite integral value. The indefinite integral value represents the result of integrating a function f(x) with respect to the variable $$$x$$$. It is such function $$$F(x)$$$ that $$$F^{\prime}(x)=f(x)$$$. The definite integral value is the value of the integral over a specified interval. It can be a numerical value or some expression. The definite integral value provides information about quantities like area, accumulated total, or displacement, depending on the context of the problem.
