Question

Evaluate the integral \( \int \frac{dx}{x(x^4 + 1)} \):

• A. \(\log\left(\frac{x}{x^4 + 1}\right) + C\)
• B. \(\frac{3}{4} \log(x^4 + 1) + C\)
• C. \(\frac{1}{3} \log\left(\frac{x^3}{x^4 + 1}\right) + C\)
• D. \(\frac{1}{4} \log\left(\frac{x^4}{x^4 + 1}\right) + C\)

Hint:

When dealing with integrals involving complex fractions, partial fraction decomposition and substitution can significantly simplify the process. Always check if the integrand can be split into simpler terms for easier integration.

Answer 1

The Correct Option is D

We are tasked with evaluating the integral: \[ I = \int \frac{dx}{x(x^4 + 1)}. \] To solve this, first notice the structure of the denominator, which suggests a decomposition or substitution. 
Step 1: Perform a partial fraction decomposition. We decompose the fraction into two simpler fractions: \[ \frac{1}{x(x^4 + 1)} = \frac{A}{x} + \frac{Bx^3 + Cx}{x^4 + 1}. \] Multiplying through by \(x(x^4 + 1)\) to clear the denominator: \[ 1 = A(x^4 + 1) + (Bx^3 + Cx)x. \] This simplifies to: \[ 1 = A(x^4 + 1) + Bx^4 + Cx^2. \] Now, collect like terms: \[ 1 = (A + B)x^4 + Cx^2 + A. \] For this to hold for all \(x\), the coefficients of \(x^4\), \(x^2\), and the constant term must match. This gives the system of equations: \[ A + B = 0, \quad C = 0, \quad A = 1. \] Solving this system: \[ A = 1, \quad B = -1, \quad C = 0. \] Thus, we have: \[ \frac{1}{x(x^4 + 1)} = \frac{1}{x} - \frac{x^3}{x^4 + 1}. \] Step 2: Integrate each term separately. Now, the integral becomes: \[ I = \int \frac{1}{x} dx - \int \frac{x^3}{x^4 + 1} dx. \] - The first integral is straightforward: \[ \int \frac{1}{x} dx = \ln|x|. \] - For the second integral, use the substitution \( u = x^4 + 1 \), so \( du = 4x^3 dx \), or \( \frac{du}{4} = x^3 dx \): \[ \int \frac{x^3}{x^4 + 1} dx = \frac{1}{4} \int \frac{du}{u} = \frac{1}{4} \ln|u| = \frac{1}{4} \ln(x^4 + 1). \] Step 3: Combine the results. Thus, the total integral is: \[ I = \ln|x| - \frac{1}{4} \ln(x^4 + 1) + C. \] Step 4: Express the final answer. We can combine the logarithmic terms: \[ I = \frac{1}{4} \ln\left(\frac{x^4}{x^4 + 1}\right) + C. \] Thus, the correct answer is: \[ \boxed{\frac{1}{4} \log\left(\frac{x^4}{x^4 + 1}\right) + C}. \]