Integrals by Partial Fraction Expansion Calculator

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Example

Created on 2024-06-20Asked by William Nelson (Solvelet student)

Evaluate the integral

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

using partial fraction expansion.

Solution

To evaluate the integral

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

using partial fraction expansion: First, factor the denominator:

x^2 + x = x(x + 1)

Next, express the integrand as a sum of partial fractions:

\frac{3x+2}{x(x+1)} = \frac{A}{x} + \frac{B}{x+1}

Clear the denominators by multiplying both sides by

x(x + 1)

3x + 2 = A(x + 1) + Bx

Expand and collect like terms:

3x + 2 = Ax + A + Bx

Equating coefficients of like terms:

A + B = 3

(coefficients of

x

)

A = 2

(constant terms) Solve the system of equations to find

A

and

B

A = 2

2 + B = 3

B = 1

Now, rewrite the original integral with the partial fractions:

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

Integrate each term separately:

\int \frac{2}{x} \, dx = 2 \ln|x|

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

Therefore, the integral

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

is:

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

Where

C

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

Some of the related questions asked by Theodore Nguyen on Solvelet

1. Decompose the rational function

f(x) = \frac{2x + 3}{x^2 - x - 6}

into partial fractions,2. Find the area of the rectangle with base

6

and height

4

using integration.,

DefinitionPartial fraction expansionPartial fraction expansion is useful when we want to calculate an integral of a rational function. This method reduces the complexity in integration as it expresses the big expressions as sums of smaller rational functions. It is especially effective when integrating rational functions if the degree of the numerator is less than the degree of the denominator. For example, ∫(x−1)(x+2)2x+3dx can be decomposed into ∫(x−1A+x+2B)dx and integrated from there.

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