Integrate (x^2 + 3) / ((x^2 - 1)(x^2 - 2)) dx

Steps to Solve

1. Set Up the Integral

We start with the integral we need to solve:

$$ \int \frac{x^2 + 3}{(x^2 - 1)(x^2 - 2)} , dx $$

2. Perform Partial Fraction Decomposition

We'll express the integrand as the sum of partial fractions:

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

Multiplying both sides by ((x^2 - 1)(x^2 - 2)) gives:

$$ x^2 + 3 = A(x^2 - 2) + B(x^2 - 1) $$

3. Expand and Collect Terms

Expanding the right side:

$$ x^2 + 3 = Ax^2 - 2A + Bx^2 - B $$

Combine like terms:

$$ x^2 + 3 = (A + B)x^2 + (-2A - B) $$

4. Set Up the System of Equations

Now we can set up our equations based on coefficients:

1. ( A + B = 1 )

2. ( -2A - B = 3 )

3. Solve for A and B

From the first equation, express ( B ) in terms of ( A ):

$$ B = 1 - A $$

Substituting in the second equation:

$$ -2A - (1 - A) = 3 $$

This simplifies to:

$$ -2A - 1 + A = 3 $$

Combine like terms:

$$ -A - 1 = 3 $$

Solving for ( A ):

$$ -A = 4 \implies A = -4 $$

Now substitute ( A ) back into the equation for ( B ):

$$ B = 1 - (-4) = 5 $$

6. Rewrite the Integral with Partial Fractions

Now we can rewrite the integral:

$$ \int \left( \frac{-4}{x^2 - 1} + \frac{5}{x^2 - 2} \right) , dx $$

7. Integrate Each Term

We can now integrate each term separately:

For ( \frac{-4}{x^2 - 1} ):

$$ \int \frac{-4}{x^2 - 1} , dx = -4 \int \frac{1}{x^2 - 1} , dx $$

For ( \frac{5}{x^2 - 2} ):

$$ \int \frac{5}{x^2 - 2} , dx = 5 \int \frac{1}{x^2 - 2} , dx $$

8. Integral Results

The integrals result in:

1. ( -4 \cdot \frac{1}{2} \ln \left| \frac{x-1}{x+1} \right| + C_1 = -2 \ln \left| \frac{x-1}{x+1} \right| + C_1 )

2. ( 5 \cdot \frac{1}{\sqrt{2}} \tan^{-1} \left( \frac{x}{\sqrt{2}} \right) + C_2 )

Thus, the final result of the integration is:

$$ -2 \ln \left| \frac{x-1}{x+1} \right| + \frac{5}{\sqrt{2}} \tan^{-1} \left( \frac{x}{\sqrt{2}} \right) + C $$