Question

Evaluate: \[ \int \frac{1}{x^2 + 25} dx. \]

Hint:

The integral \( \int \frac{1}{x^2 + a^2} \, dx \) is evaluated using: \[ \frac{1}{a} \tan^{-1} \left(\frac{x}{a}\right) + C. \]

Answer 1

Step 1: Recognizing the Standard Integral Formula
We apply the standard integral formula: \[ \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1} \left(\frac{x}{a}\right) + C. \] Step 2: Substituting \( a = 5 \)
Given that \( a^2 = 25 \), we find \( a = 5 \). Thus, the integral becomes: \[ \int \frac{1}{x^2 + 25} \, dx = \frac{1}{5} \tan^{-1} \left(\frac{x}{5}\right) + C. \]