Questions: Use the Log Rule to find the indefinite integral. [ int fracx+4x^2+8 x+1 d x ]

Solution Steps

To solve the integral \(\int \frac{x+4}{x^{2}+8x+1} \, dx\), we can use the method of partial fraction decomposition or substitution if applicable. However, in this case, the denominator does not factor easily, and the numerator is a linear polynomial. We can attempt a substitution where the derivative of the denominator is related to the numerator. Let \( u = x^2 + 8x + 1 \), then \( du = (2x + 8) \, dx \). We can adjust the numerator to fit this form and proceed with the integration.

We start with the integral

\[ \int \frac{x+4}{x^{2}+8x+1} \, dx. \]

Let

\[ u = x^2 + 8x + 1. \]

Then, the derivative is

\[ du = (2x + 8) \, dx. \]

We can express \(dx\) in terms of \(du\):

\[ dx = \frac{du}{2x + 8}. \]

We can rewrite the integral in terms of \(u\). The numerator \(x + 4\) can be adjusted to fit the form of \(du\):

\[ \int \frac{x + 4}{u} \, dx = \int \frac{x + 4}{u} \cdot \frac{du}{2x + 8}. \]

After simplifying, we find that the integral evaluates to:

\[ \frac{1}{2} \log(u) + C = \frac{1}{2} \log(x^2 + 8x + 1) + C. \]

Final Answer