Question

Evaluate the following integrals: \[ \int \frac{(x^4 + 1)}{x(2x + 1)^2} \, dx \] and \[ \int \frac{1}{x^4 + 5x^2 + 6} \, dx \]

• A. \( \frac{1}{(2x + 1)} \)
• B. \( \frac{1}{(x^4 + 5x^2 + 6)} \)
• C. \( \frac{1}{2} \left( \ln \left| \frac{x^2 + 3}{x + 2} \right| \right) \)
• D. \( \frac{1}{3} \left( \ln |x^2 + 5x + 6| \right) \)

Hint:

When dealing with integrals of rational functions, look for opportunities to simplify and split into manageable parts, and use substitution when applicable.

Answer 1

The Correct Option is C

This task involves evaluating two integrals: one rational and one simple algebraic. We begin with the first integral.
Step 1: Solution of the first integralThe given integral is:\[\int \frac{(x^4 + 1)}{x(2x + 1)^2} \, dx\]To simplify, we decompose the expression:\[\frac{x^4 + 1}{x(2x + 1)^2} = \frac{x^4}{x(2x + 1)^2} + \frac{1}{x(2x + 1)^2}\]This results in two separate integrals:\[\int \frac{x^4}{x(2x + 1)^2} \, dx + \int \frac{1}{x(2x + 1)^2} \, dx\]The first term simplifies to:\[\int \frac{x^4}{x(2x + 1)^2} \, dx = \int \frac{x^3}{(2x + 1)^2} \, dx\]For the second integral, a substitution \( u = 2x + 1 \), with \( du = 2dx \), can be applied for easier integration.
Step 2: Solution of the second integralWe now address the second integral:\[\int \frac{1}{x(2x + 1)^2} \, dx\]
Step 3: Result EvaluationThrough algebraic simplification and integration, the evaluated expression is:\[\boxed{\frac{1}{2} \ln \left| \frac{x^2 + 3}{x + 2} \right|}\]Therefore, the solution for the first integral is \( \frac{1}{2} \ln \left| \frac{x^2 + 3}{x + 2} \right| \).