Question

Evaluate: \( \int \frac{1}{25 - 9x^2} \, dx \)

Hint:

For integrals of the form \( \frac{1}{a^2 - x^2} \), use the standard formula involving natural logarithms.

Answer 1

Step 1: Simplify the integrand.
The integrand is in the form \( \frac{1}{a^2 - x^2} \), which is a standard form for integration. We use the formula: \[ \int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln \left| \frac{a + x}{a - x} \right| + C \] Here, \( a^2 = 25 \), so \( a = 5 \).

Step 2: Apply the formula.
Substitute \( a = 5 \) and \( x \) into the standard formula: \[ \int \frac{dx}{25 - 9x^2} = \frac{1}{2 \cdot 5} \ln \left| \frac{5 + 3x}{5 - 3x} \right| + C \] \[ = \frac{1}{10} \ln \left| \frac{5 + 3x}{5 - 3x} \right| + C \]

Final Answer: \[ \boxed{\frac{1}{10} \ln \left| \frac{5 + 3x}{5 - 3x} \right| + C} \]