Integrals of Rational Functions Calculator

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Example

Created on 2024-06-20Asked by Victoria Robinson (Solvelet student)

Evaluate the integral

\int \frac{x^2 + 3x + 2}{x + 1} \, dx

Solution

To evaluate the integral

\int \frac{x^2 + 3x + 2}{x + 1} \, dx

: Perform long division or use partial fraction decomposition to rewrite the integrand:

\frac{x^2 + 3x + 2}{x + 1} = x + 2 + \frac{0}{x + 1}

Now, integrate each term separately:

\int (x + 2) \, dx + \int \frac{0}{x + 1} \, dx

Use the power rule for integration:

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C

Apply the power rule to each term:

\int (x + 2) \, dx = \frac{x^2}{2} + 2x + C

\int \frac{0}{x + 1} \, dx = 0 + C

Combine the results:

\int \frac{x^2 + 3x + 2}{x + 1} \, dx = \frac{x^2}{2} + 2x + C

Therefore, the integral

\int \frac{x^2 + 3x + 2}{x + 1} \, dx

is:

\int \frac{x^2 + 3x + 2}{x + 1} \, dx = \frac{x^2}{2} + 2x + C

Where

C

is the constant of integration. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Levi Carter on Solvelet

1. Evaluate the integral

\int_{-2}^2 \frac{x + 2}{x^2 - 4} \, dx

,2. Find the period of the function

f(x) = \frac{\sin(2x)}{1 + \cos(x)}

using integration.,

DefinitionRational functions, that are ratios of polynomials, are integrable in certain conditions by use of partial fraction decomposition, substitution, or contour integration. These methods convert these complicated rational functions to the form that is easier to integrate. Rational function integrals are important in advanced calculus, control theory, and signal processing applications. One common example: The integral of ∫(x=2 + 1)/x dx is solved through a partial fraction that leads to the integral of ∫(1/2) du = 1/2 [ln x2 + 1].

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