## Evaluating Integrals

An Integration calculator allows you to learn the concepts of calculating integrals without spending too much time. You can evaluate the integral using an integral calculator with steps easily online.

Similarly, you can find a double integral calculator on this website. The double integral calculator shows you graphs, plots, steps, and visual representation, which helps you learn advanced concepts of double integration.

There are many other useful calculators you can use to get benefit. Similarly, you can determine the volume of a solid of revolution with a washer method calculator and determine the cross sections of a solid of revolution with a disc method calculator.

## How to use the Integral calculator?

Using the integration by parts calculator is easy and fast.

Follow these steps:

### Step 1: Enter the function

To evaluate the integrals, you must have a proper function. You need to enter your function in the function bar of the integration calculator. There is also a "load example" list. You can click that list to load an example equation for calculating integrals step by step.

### Step 2: Select the Variable

For evaluating integrals, there are three variables you can use. These variables are x, y and z. The role of these three variables differs from each other, and all 3 have different impacts on the overall output. You can select the variables as x, y and z from the variable section.

### Step 3: Give Upper bound value

The upper bound is the value that helps us sum integral at its maximum value. The upper bound is denoted as U, and its determination is crucial in the integration process. You can enter the upper bound of your limit in the upper bound section of the upper bound calculator.

### Step 4: Give Lower bound value

The lower bound is the smallest value that we set to start the integration. To get the accurate integration results, the smallest value of an interval is denoted by L. To get the precise integration results. You need to enter the actual amount of your lower bound limit in the lower bound section of the upper and lower bounds calculator.

After completing all above steps Press "GO" button.

Immediately after clicking the button our integral calculus calculator will start working. The integration by parts calculator will show you the anti derivative, integral steps, parsing tree and plot of your result. All these functionalities and features makes this the best line integral calculator to evaluate the integral of complicated integration problems.

## What are integrals

An **integral** in calculus refers to a fundamental concept that is essentially the reverse process of differentiation. There are two key types of integrals we teach our students: definite and indefinite.

### 1. Indefinite Integral (Antiderivative):

The indefinite integral of a function represents a family of functions whose derivatives are the given function. Mathematically, it is represented as:

\[ \int f(x) \, dx = F(x) + C \]

where \( F(x) \) is an antiderivative of \( f(x) \) and \( C \) is an arbitrary constant.

### 2. Definite Integral:

The definite integral of a function over a specific interval \([a, b]\) calculates the net area between the function and the x-axis within that interval. This area is accounted for in both directions, meaning areas above the x-axis are treated as positive, while those below are treated as negative. It's mathematically represented as:

\[ \int_a^b f(x) \, dx \]

In layman's terms:

- Integrals can be visualized as the
.**area under**a curve - They represent an
or total of quantities. For instance, integrating a velocity function gives a distance (or displacement) function.*accumulation* - In physics, integrals are used in diverse scenarios such as calculating work done by a force or the distribution of charges in electric fields.

The process of finding integrals is called **integration**.

## Understanding Integration

To solve for a definite integral, you have to understand first that definite integrals have start and endpoints, also known as limits or intervals, represented as (a,b) and are placed on top and bottom of the integral.

We can generalize integrals based on functions and domains through which integration is done. Integration by parts calculator with steps helps you to evaluate the integrals digitally.

**Also: **You can find the Line Integral Calculator and Surface Integral Calculator for more Information.

The formula for integral (definite) goes like this:

$\int_b^a f(x)dx=-\int_b^a f(x)dx$

$\int_b^a f(x)dx$

Where,

∫ represents integral

dx represents the differential of the 'x' variable

fx represents the integrand

point a and b represent limits of integration

Let's solve it considering that we're being asked for integral from 1 to 3, of 3x dx

$\int_3^1 3(x)dx$

Solving:

$\int_b^a f(x)dx=-\int_b^a f(x)dx$

$-\int_3^1 3(x)dx$

Take out the constant:

$\int a.f(x)dx = a.\int f(x)dx$

Applying the power rule:

$\int x^a dx = x^a+1/a+1, a≠1$

$=-3[\frac{x^{1+1}}{1+1}]_1^3$

simplify

$-3[\frac{x^2}{2}]_1^3$

calculating the boundaries: 4

-3.4

-12

On the other hand, the Indefinite integral is distinguished from the definite integral because of the former’s lack of defined limits.

**Indefinite Integral goes by the formula:**

$\int f(x)dx$

The above integration solver can calculate indefinite integral and definite integral, but if you want to calculate indefinite integral only, find the best online indefinite integral calculator.

**Related:** Learn about definite integral and indefinite integral

## How to calculate improper Integral?

One of the reasons why a definite integral becomes an improper integral is when one or both of the limits reach infinity. An Integral calculus calculator can be used to calculate improper integrals.

This integral is then solved by turning it into a problem of limits where c happens to approach infinity or negative infinity.

Let's consider an example where one of the limits of integration is infinite and then solve it.

$\int_1^\infty \frac{1}{x^2} dx \;and\; \int_1^\infty \frac{1}{x} dx$

$\int_1^\infty \frac{1}{x^2} dx = \lim_{c\to \infty} \int_1^c \frac{1}{x^2} dx$

$\lim_{c\to \infty} [-\frac{1}{x}]_1^c$

$\lim_{c\to \infty} [-\frac {1}{c}] -(-\frac{1}{1})]$

0+1

1

Since the answer to the improper integral is finite, we consider it converged.

If you only want to evaluate definite integrals, use this best step by step definite integral calculator online.

**Related:** Use shell method calculator with steps to find the volume of a solid of revolution easily online.

## How to calculate Continuous Integration?

The fundamental theorem of calculus establishes a clear association between integral and differential calculus. Our integral calculator with steps is capable enough to calculate continuous integration.

If f(x) is continuous for the interval a and b given the variable x and G(x) is a function in such sense that dG/dx = f(x) for all values of x in (a,b)

Let f be continuous on an interval ‘y’. Select a point p in y then the function f(x) is defined as:

Let F(x) be as follows

$\int_p^x f(t) dt$

Let c be in i and let x be infinitely close to c and the endpoints of i. Then by addition,

$\int_p^c f(t) dt = \int_p^x f(t) dt + \int_x^c f(t)dt$

$\int_p^c f(t) dt - \int_p^x f(t) dt + \int_x^c f(t)dt$

$f(c) - F(x)= \int_x^c f(t)dt$

For your ease and advance learning regarding multiple integrals, we offer one of the quickest triple integral calculator. This tool will surely assist you in calculating definite and indefinite triple integrals online by doing few clicks.

**Related:** Understanding Integration by Partial Fraction in 5 Minutes!

## Learning about integral calculations

### How to Evaluate Integrals?

There are two types of integrals, definite and indefinite integrals. You can solve them both by integration. The difference is that you need to put the limit values after integration in definite integrals, whereas in indefinite integrals, you don’t need to put limit values.

Integrals calculator helps to solve out every type of definite and indefinite problems easily.

### What is the integral of ex?

The integral of ex is:

$ \int e^x dx \;=\; \frac{e^x}{1}+c $ $ \int e^x dx \;=\; e^x+c$

The integration of exponential functions is tricky, but we provide great tools to evaluate integral online.

### How to Evaluate the Integral by Interpreting it in terms of Areas?

Solving integration online are basically finding the area under a specific curve. For example of a given equation of curve 1-x with upper and lower bound x=-4 and x=3, the area will be calculated as,

$ \int_{-4}^3 (1-x) dx \;=\; \Biggr| x - \frac{x^2}{2} \Biggr|_{-4}^3 $ $ \int_{-4}^3 (1-x) dx \;=\; \left( 3 - \frac{3^2}{2} \right) \;-\; \left( -4 - \frac{(-4)^2}{2} \right) $ $ \int_{-4}^3 (1-x) dx \;=\; \left( 3 - \frac{9}{2} \right) \;-\; \left( -4 -2 \right) \;=\; \frac{21}{2} $

So the area under the given curve is 21/2. We can verify this by evaluate the integral calculator for cross-checking your answer.

The calculator integral is the great resource for this type of calculations to save your time.

### What is the integral of 1/x?

The integral of 1/x is,

$ \int \frac{1}{x} dx \;=\; ln(x) + c$

## Integral Examples & Functions

Function | Integral | Explanation |
---|---|---|

1/x | ln |x| + C | The integral of 1/x with respect to x is the natural logarithm of the absolute value of x. |

2^x | 2^x/ln(2) + C | Using the formula for a^x with a = 2. |

2x | x^2 + C | Using the power rule for integration. |

x | x^2/2 + C | Applying the power rule for integration. |

e^x | e^x + C | The exponential function e^x has the same derivative and antiderivative. |

1/e^x or e^-x | -e^-x + C | Using the antiderivative for e^x, we adjust for the negative exponent. |

1 | x + C | The antiderivative of a constant is the variable of integration times that constant. |

x^x | Non-integrable function | Does not have a standard antiderivative in terms of elementary functions. |

a^x (where a is a constant) | a^x/ln(a) + C | This formula generalizes the integral for any base a. |

1/x^2 or x^-2 | -1/x + C | Applying the power rule for integration to negative powers. |

ln(x) | xln(x) - x + C | Requires integration by parts, a standard calculus technique. |

sec^2(x) | tan(x) + C | This is the derivative of the tangent function. |

## Integration terms and concepts

**Function**: A relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.**Limit**: The value that a function approaches as the input approaches a certain value.**Continuous Function**: A function for which small changes in the input result in small changes in the output, meaning it has no gaps, jumps, or holes.**Antiderivative**: A function whose derivative gives the original function. It's the reverse process of differentiation.**Indefinite Integral**: The antiderivative of a function, representing a family of functions.**Definite Integral**: Represents the signed area under the curve of a function between two limits.**Integrand**: The function being integrated.**Limits of Integration**: The values`a`

and`b`

in a definite integral, representing the interval over which the function is integrated.**Constant of Integration**: When determining the indefinite integral, there's an arbitrary constant`C`

that's added because any constant's derivative is zero.**Fundamental Theorem of Calculus**: Connects differentiation and integration. It states that if a function is continuous over an interval`[a, b]`

and`F`

is an antiderivative of`f`

on`[a, b]`

, then:**Integration by Substitution**: A method used to transform a complicated integrand into a simpler one.**Integration by Parts**: A technique based on the product rule of differentiation to integrate products of functions.**Partial Fractions**: Decomposing rational functions to simpler fractions, making them easier to integrate.**Improper Integral**: Integrals where one or both of the limits of integration are infinite, or where the integrand is unbounded.**Riemann Sum**: An approximation of the definite integral using rectangles.**Disk/Washer Method**: Techniques to find the volume of solids of revolution.**Cylindrical Shells Method**: Another method to compute the volume of solids of revolution.**Integration of Trigonometric Functions**: Special techniques and identities for integrating functions like sin(x), cos(x), tan(x), etc.**Integration of Exponential and Logarithmic Functions**: Methods and formulas to integrate functions like e^x, ln(x), etc.**Integration of Hyperbolic Functions**: Techniques for integrating functions like sinh(x), cosh(x), etc.**Linearity of the Integral**: The principle that the integral of a sum is the sum of the integrals.**Sequences and Series**: For integrating functions represented as power series or when using techniques like integration by Taylor series.**Convergence**: A property that determines whether a sequence or series approaches a finite value.

## When to use different integral calculators?

One of the biggest challeges is understanding when in how to calculate integrals. Each calculator is tailored to a specific kind of problem, and in my experience, students need to become as good at understanding what calculator to use as they are with solving integrals. Here's how I help my students understand which calculators to use:

**Integral Calculator****Use When**: You want to evaluate general integrals, either definite or indefinite.**Purpose**: Calculate the area under a curve, find antiderivatives, or accumulate quantities. Basic, go-to calculator for most problems in the classroom

**Use When**: You want to evaluate integrals over a two-dimensional region.**Purpose**: Often used to find areas, volumes, or other quantities in multivariable calculus. A bit more advanced for my students!

**Use When**: You're integrating over a three-dimensional region.**Purpose**: Useful for determining volumes and other three-dimensional quantities in multivariable calculus. This is really advanced. Used in engineering more than the classroom.

**Use When**: You want to find the net area under a curve between two specific points.**Purpose**: To calculate a specific value representing the area between the function and the x-axis over a given interval.

**Indefinite Integral Calculator****Use When**: You want to find the general antiderivative of a function without specific bounds.**Purpose**: Outputs a function (or family of functions) representing the integral, typically including a constant of integration.

**Use When**: You're finding the volume of a solid of revolution about a non-central axis using cylindrical shells.**Purpose**: It's a technique for volume when the disc/washer method is not applicable. You'll have to use your best understanding to know when the Shell method is appropriate

**Use When**: You're determining the volume of a solid of revolution about an axis using washers (annular discs).**Purpose**: Used when the solid has a hole in the middle, like a donut shape.

**Use When**: You aim to find the volume of a solid of revolution about an axis using discs.**Purpose**: Useful for determining volumes of solids with no holes in them, like a cylinder or cone.

**Use When**: You want to transform a given function from the time domain to the frequency (s) domain.**Purpose**: Laplace Transforms are instrumental in solving linear ordinary differential equations, analyzing circuits, and other applications in engineering and physics

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