Long Division Integral Calculator

Discover the efficiency of seamless integration with our long division integral calculator, designed to simplify complex rational functions effortlessly.

Definite Integral
Indefinite Integral

with respect to

Upper Bound

Lower Bound

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Introduction to Long Division Integral Calculator:

The long division integral calculator is a great online tool that can simplify the process of integrating rational functions. This tool helps in evaluating polynomials where the numerator polynomial is greater than or equal to the degree of the denominator polynomial.

Long Division Integral Calculator with Steps

It is a helpful tool as it gives accurate results to students and professionals who have to complete tasks in mathematics and related fields.

What is the Long Division Integral?

Long division integral is a method in calculus that is used to divide the numerator polynomial by the denominator polynomial to integrate the given expression easily. The result of this process is a quotient and a remainder, allowing the rational function integral to be broken down into simpler parts.

There is another method called partial fraction method, used to solve the rational function problems but for complex rational integral functions, the integration method is more suitable.

How to Integrate Using Long Division?

Long division in integration simplifies the process of integrating rational functions by breaking them down into simpler ones. Let us see an example of long-division integration:

Example: Integrate the following:

$$ \frac{x^3 + 2x^2 + x + 1}{x^2 + 1} $$

Solution:

Step 1:

Write the numerator polynomial inside the long division symbol and the denominator polynomial outside.

$$ x^2 + 1 \sqrt{x^3 + 2x^2 + x + 1} $$

Step 2:

Divide the leading term of the numerator by the leading term of the denominator to get the first term of the quotient by multiplying the entire denominator with this term, and subtracting the result from the numerator.

\begin{array}{rrrrrrr} x + 2 \\[-3pt] x^2 + 1 \enclose{longdiv}{x^3+ 2x^2 + x + 1} \\[-3pt] \underline{-x^3 \; \; \; \; \; \; \; \; \; \; \pm x\; \; \; \; \;}\phantom{2} \\[-3pt] 2x^2 \; \; \; \; \; \; \; \; \; + 1 \\[-3pt] \underline{-2x^2\; \; \; \; \; \; \; \; \; \pm 2} \\[-3pt] -1 \end{array}

In this way, the long division process continues, until the remainder is not less than the denominator value.

Step 3:

After division, rewrite the original rational function as the sum of the polynomial quotient and the remainder divided by the original denominator.

$$ \frac{x^3 + 2x^2 + x + 1}{x^2 + 1} \;=\; x + 2 + \frac{-1}{x^2 + 1} $$

Step 4:

Now you can use the partial fraction method to simplify more to make the integration process smooth.

How to Use the Integral Long Division Calculator?

Long division integration calculator has a simple interface, so you can easily use it to evaluate the integral long division problem solution. Before adding the input for the solutions of given long division problems, you must follow some steps.

Enter the numerator value of the rational function in the input field (that you want to evaluate using the long division method).
Enter the denominator value of the rational function in the second input field.
Recheck your input value for the long division integral problem solution before starting the calculation in long division integral calculator.
Click on the “Calculate” button to get the desired result of your given long division integral problem.
If you are trying our long division integral solver for the first time then use the load example option.
Click on the “Recalculate” button to get a new page for solving more long division integral questions.

Final Result of Long Division Integration Calculator:

Integral long division calculator gives you the solution of rational function when you give it an input value. It provides you with solutions that contain as:

Result Option:

Click on the result option and it will provide you with a solution of long division integral questions.

Possible Step:

When you click on the possible steps option it will provide you with the step by step solution of the long division integral problem.

Advantages of Long Division Integral Calculator:

Long division integral solver gives you many benefits that you get whenever you use it to calculate rational functions for integration. These benefits are:

Our tool saves the time and effort that you consume in solving complex long division integral questions and gives solutions in a few seconds.
Long division integration calculator is a free-of-cost tool that provides a solution for a long division integral problem and simplify the integration process for free.
This tool ensures that you get the correct solution when you are computing lengthy rational integral function problem.
Integral long division calculator is an adaptive tool that allows you to find the long division integral function.
You can use this calculator for practicing different concepts of long division and solve multiple examples.
Long division integral calculator is a trustworthy tool that provides you with correct solutions of long division integral problem as per your input.

Related References

What is Long Division Integration?
Give some examples of long divison integration:
How to Evaluate long division integration?
Explain long division integral with the help of examples.

Frequently Ask Questions

When to use long division in integration

Long division is an appropriate method in integration for simplifying the process of integrating rational functions, which are ratios of two polynomials.

You can use the long division when the degree of the numerator polynomial P(x) is greater than or equal to the degree of the denominator polynomial Q(x).

If the degree of P(x) is less than the degree of Q(x), long division is not needed, and you can use other methods of integration, such as partial fraction decomposition or direct integration.

When you have a rational function where direct integration is complicated by the high degree of the numerator, the long division helps to simplify the integrand into a polynomial.

When using Integration by Partial Fractions, do you have to use Long Division?

When using partial fractions for integration, you need to use long division only if the degree of the numerator is greater than or equal to the degree of the denominator. If the degree of the numerator is less than the degree of the denominator, you can directly proceed with the partial fraction decomposition without using long division.

To evaluate the indefinite integral of (7x+3)/(x+8), what is the first step?

As you can see the degree of a polynomial of both numerator or denominator is equal so use the long division method such as,

\begin{array}{r} 7 \\[-3pt] x + 8 \enclose{longdiv}{7x + 3} \\[-3pt] \underline{-7x \pm 56}\phantom{2} \\[-3pt] -53 \\[-3pt] \end{array}

$$ \frac{(7x + 3)}{x+8} \;=\; 7 + \frac{-53}{x+8} $$

Now it becomes the simplest expression so use the direct integral method for integration,

$$ \int 7 dx \;=\; 7x + C_1 $$

$$ \int \frac{53}{x+8} dx \;=\; 53 \int \frac{1}{x+8} dx $$

$$ u \;=\; x + 8,\; hence\; du \;=\; dx $$

$$ \int \frac{1}{x+8} dx \;=\; \int \frac{1}{u} du \;=\; ln |u| + C_2 \;=\; ln |x + 8| + C_2 $$

Combine the integral and the solution of the given expression is,

$$ \int (7 - \frac{53}{x+8}) dx \;=\; 7x - 53 ln |x + 8| + C $$

Integrate (x ^ 2 + 3x ^ 2 + 5)/(x - 3) dx

In the given question degree of polynomial of numerator is greater than denominator so use the long division method. The numerator 4x^22+5 is of degree 2 and the denominator x-3 is of degree 1. So first do the long division,

\begin{array}{r} 4x + 12 \\[-3pt] x - 3 \enclose{longdiv}{4x^2 + 0x + 5} \\[-3pt] \underline{-4x^2 \mp 12x}\phantom{2} \\[-3pt] 12x + 5 \\[-3pt] \underline{-12x \pm 36} \phantom{2} \\[-3pt] 41 \end{array}

$$ \left(\frac{x^2 + 3x^2 + 5}{x-3} \right) \;=\; 4x + 12 + \frac{41}{x - 3} $$

Now the given expression is in simplify form so you can easily integrate it using direct rule of integration.

Integrate 4x with respect to x,

$$ \int 4x dx \;=\; 2x^2 + C_1 $$

Integrate 12 with respect to x,

$$ \int 12 dx \;=\; 12x + C_2 $$

Integrate 41/x-3 with respect to x,

$$ \int \frac{41}{x-3} dx \;=\; 41 \int \frac{1}{x-3} dx $$

$$ \int \frac{1}{x-3} dx \;=\; ln |x - 3| + C_3 $$

Combine all the values to get the solution of given integral function,

$$ \int (4x + 12 + \frac{41}{x-3}) dx \;=\; 2x^2 + 12x + 41 ln |x - 3| + C $$

Amanda Jepson

Last Updated February 21, 2024