$\int_{0}^{\pi}\frac{x}{a^{2}cos^{2}x+b^{2}sin^{2}x}dx$ is equal to:

Question

Accepted Answer

To solve the integral $\int_{0}^{\pi}\frac{x}{a^{2}\cos^{2}x+b^{2}\sin^{2}x}dx$, we need to evaluate it using appropriate techniques and verify why Option 2, $\frac{\pi^{2}}{2ab}$, is the correct answer.

### Explanation for Option 2:

1. **Symmetry and Substitution:**
   The integral has limits from 0 to $\pi$, which suggests that symmetry might play a role. Consider the substitution $x = \pi - t$, which transforms the integral as follows:
   $$
   \int_{0}^{\pi} \frac{x}{a^{2}\cos^{2}x + b^{2}\sin^{2}x} \, dx = \int_{0}^{\pi} \frac{\pi - t}{a^{2}\sin^{2}t + b^{2}\cos^{2}t} \, dt
   $$
   This substitution exploits the symmetry of the trigonometric functions over the interval $[0, \pi]$.

2. **Combining Integrals:**
   Adding the original integral and the transformed integral, we get:
   $$
   2I = \int_{0}^{\pi} \frac{x}{a^{2}\cos^{2}x + b^{2}\sin^{2}x} \, dx + \int_{0}^{\pi} \frac{\pi - x}{a^{2}\sin^{2}x + b^{2}\cos^{2}x} \, dx
   $$
   Simplifying, this becomes:
   $$
   2I = \pi \int_{0}^{\pi} \frac{1}{a^{2}\cos^{2}x + b^{2}\sin^{2}x} \, dx
   $$

3. **Evaluating the Integral:**
   The integral $\int_{0}^{\pi} \frac{1}{a^{2}\cos^{2}x + b^{2}\sin^{2}x} \, dx$ can be evaluated using the identity for the integral of a rational function of sine and cosine. The result of this integral is $\frac{\pi}{ab}$.

4. **Final Calculation:**
   Substituting back, we have:
   $$
   2I = \pi \cdot \frac{\pi}{ab} = \frac{\pi^2}{ab}
   $$
   Therefore, $I = \frac{\pi^2}{2ab}$.

Thus, the correct answer is Option 2: $\frac{\pi^{2}}{2ab}$.

### Explanation for Why Other Options are Incorrect:

- **Option 1: $\frac{\pi^{2}}{ab}$**
  - This option represents the result of the integral without considering the factor of 2 that arises from the symmetry and substitution method. The integral $I$ is half of this value, making Option 1 incorrect.

- **Option 3: $\frac{\pi^{2}}{4ab}$**
  - This option suggests a further division by 2, which is not justified by the symmetry and substitution method used. The calculations clearly show that the factor is $\frac{1}{2}$, not $\frac{1}{4}$.

- **Option 4: $\frac{\pi^{2}}{9ab}$**
  - This option is incorrect as it suggests an additional division by 4.5, which is not supported by any step in the calculation process. The integral evaluation and symmetry considerations do not lead to this factor.

In conclusion, the correct answer is Option 2: $\frac{\pi^{2}}{2ab}$, as justified by the symmetry and substitution method used in evaluating the integral.