Question

$$ \int \frac{dx}{x(x^{10}+1)} = $$

• A. $\frac{1}{10} \log \left( \frac{x^{10}}{x^{10}+1} \right) + c$
• B. $\frac{1}{10} \log \left( \frac{x^{10}}{x^{10}-1} \right) + c$
• C. $\log \left( \frac{x^{10}}{x^{10}+1} \right) + c$
• D. None

Hint:

General shortcut: $\int \frac{dx}{x(x^n+1)} = \frac{1}{n} \log \left( \frac{x^n}{x^n+1} \right) + c$. This works for any positive integer $n$.

Answer 1

The Correct Option is A

To solve the integral given by \(\int \frac{dx}{x(x^{10}+1)}\), we will employ the method of partial fraction decomposition. The goal is to express the integrand in a simpler form that allows us to integrate it more easily.

1. First, observe that the expression can be decomposed as: \(\frac{1}{x(x^{10}+1)} = \frac{A}{x} + \frac{Bx^9 + Cx^8 + \cdots + Jx + K}{x^{10}+1}\). However, a more straightforward approach is to use a substitution due to the nature of the polynomial.
2. Let us perform the substitution \(u = x^{10}\). Consequently, the differential changes as follows: \(du = 10x^9 \, dx \implies dx = \frac{du}{10x^9}\).
3. Now, the integral becomes: \(\int \frac{1}{x(u+1)} \cdot \frac{du}{10x^9}\) which simplifies to, \(\frac{1}{10} \int \frac{du}{u(u+1)}\).
4. This can be further decomposed into partial fractions. We express: \(\frac{1}{u(u+1)} = \frac{A}{u} + \frac{B}{u+1}\). Solving this, we get \(A = 1\) and \(B = -1\).
5. Thus, the integral becomes: \(\frac{1}{10} \int \left( \frac{1}{u} - \frac{1}{u+1} \right) du\).
6. Integrating the terms separately, we have:

\(\frac{1}{10} \left( \log |u| - \log |u+1| \right) + c = \frac{1}{10} \log \left( \frac{u}{u+1} \right) + c\).

1. Substituting back \(u = x^{10}\), we obtain: \(\frac{1}{10} \log \left( \frac{x^{10}}{x^{10}+1} \right) + c\).
2. Hence, the solution to the integral is:

\(\frac{1}{10} \log \left( \frac{x^{10}}{x^{10}+1} \right) + c\).

Therefore, the correct answer is \(\frac{1}{10} \log \left( \frac{x^{10}}{x^{10}+1} \right) + c\).