Question

Find: \[ \int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, dx \]

Hint:

Quick Tip: When dealing with rational functions involving quadratics, use partial fraction decomposition to break the function into simpler integrals. This allows you to use standard integral formulas for each term.

Answer 1

To evaluate the integral: \[ \int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, dx \] We use the method of partial fractions. We begin by expressing the integrand as: \[ \frac{2x}{(x^2 + 3)(x^2 - 5)} = \frac{A}{x^2 + 3} + \frac{B}{x^2 - 5} \] Multiplying both sides by \( (x^2 + 3)(x^2 - 5) \) to clear the denominators: \[ 2x = A(x^2 - 5) + B(x^2 + 3) \] Expanding both sides: \[ 2x = A x^2 - 5A + B x^2 + 3B \] \[ 2x = (A + B) x^2 + (-5A + 3B) \] Now, equate the coefficients of \( x^2 \) and \( x \) on both sides. For the \( x^2 \)-terms: \[ A + B = 0 \] For the \( x \)-terms: \[ -5A + 3B = 2 \] Solving this system of equations: From \( A + B = 0 \), we have \( B = -A \). Substituting this into the second equation: \[ -5A + 3(-A) = 2 \] \[ -5A - 3A = 2 \] \[ -8A = 2 \] \[ A = -\frac{1}{4} \] Since \( B = -A \), we have: \[ B = \frac{1}{4} \] Thus, the partial fractions decomposition is: \[ \frac{2x}{(x^2 + 3)(x^2 - 5)} = \frac{-1/4}{x^2 + 3} + \frac{1/4}{x^2 - 5} \] Now, integrate each term: \[ \int \frac{-1/4}{x^2 + 3} \, dx = -\frac{1}{4} \int \frac{1}{x^2 + 3} \, dx = -\frac{1}{4} \cdot \frac{1}{\sqrt{3}} \tan^{-1} \left( \frac{x}{\sqrt{3}} \right) \] \[ \int \frac{1/4}{x^2 - 5} \, dx = \frac{1}{4} \int \frac{1}{x^2 - 5} \, dx = \frac{1}{4} \cdot \frac{1}{\sqrt{5}} \tanh^{-1} \left( \frac{x}{\sqrt{5}} \right) \] Thus, the integral is: \[ \int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, dx = -\frac{1}{4\sqrt{3}} \tan^{-1} \left( \frac{x}{\sqrt{3}} \right) + \frac{1}{4\sqrt{5}} \tanh^{-1} \left( \frac{x}{\sqrt{5}} \right) + C \]