Evaluate the integral $\int \dfrac{10}{x^4 - x^2} dx$

Neetesh Kumar | September 19, 2024

This is the solution to Myopenmath Math2A Differential Equation
Integral by Partial Fraction homework question
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Step-by-Step Solution:

Step 1: Factor the Denominator

We start with the integral:

$\int \frac{10}{x^4 - x^2} \, dx$

First, we factor the denominator:

$x^4 - x^2 = x^2(x^2 - 1) = x^2(x - 1)(x + 1)$

Thus, the integral becomes:

$\int \frac{10}{x^2(x - 1)(x + 1)} \, dx$

Step 2: Partial Fraction Decomposition

We express the integrand as a sum of partial fractions:

$\frac{10}{x^2(x - 1)(x + 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 1} + \frac{D}{x + 1}$

Multiplying both sides by the denominator $x^2(x - 1)(x + 1)$, we get:

$10 = A x(x - 1)(x + 1) + B(x - 1)(x + 1) + Cx^2(x + 1) + Dx^2(x - 1)$

We will now solve for $A$, $B$, $C$, and $D$ by substituting appropriate values for $x$.

Step 3: Solve for the Coefficients

The coefficients near the like terms should be equal, so the following system is obtained:

$\begin{cases} A+C+D = 0 \\ B-C+D = 0 \\ -B = 10 \end{cases}$

Now, after solving the above system of Linear Equations, we get A = 0, B = -10, C = -5, D = 0

You can use this System of Linear Equations Calculator

Thus, we have the following decomposition:

$\frac{10}{x^2(x - 1)(x + 1)} = -\frac{10}{x^2} + \frac{5}{x - 1} - \frac{5}{x + 1}$

Step 4: Integrate Each Term

We now integrate each term separately:

1. $\int \frac{-10}{x^2} \, dx = \frac{10}{x}$
2. $\int \frac{5}{x - 1} \, dx = 5 \ln|x - 1|$
3. $-\int \frac{5}{x + 1} \, dx = -5 \ln|x + 1|$

Step 5: Combine the Results

Thus, the solution to the integral is:

$\frac{10}{x} + 5 \ln|x - 1| - 5 \ln|x + 1| + C$

Final Answer

$\boxed{\frac{10}{x} + 5 \ln|x - 1| - 5 \ln|x + 1| + C}$

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