Question

Evaluate: \[ \int_0^\pi \frac{dx}{a^2 \cos^2 x + b^2 \sin^2 x}. \]

Hint:

For integrals involving trigonometric functions, using standard integral formulas can help simplify the calculation process.

Answer 1

The integral can be evaluated using standard integration techniques. First, constants are factored out:\[I = \int_0^\pi \frac{dx}{a^2 \cos^2 x + b^2 \sin^2 x}.\]The standard formula for integrals of the form \( \int \frac{dx}{A \cos^2 x + B \sin^2 x} \) is applied. The result for this integral is:\[I = \frac{\pi}{\sqrt{a^2 + b^2}}.\]Therefore, the evaluated result is:\[\int_0^\pi \frac{dx}{a^2 \cos^2 x + b^2 \sin^2 x} = \frac{\pi}{\sqrt{a^2 + b^2}}.\]