Question

The value of the definite integral \( \int_0^{\pi} \sin^2 x \, dx \) is:

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{6} \)

Hint:

When dealing with integrals of trigonometric functions like \( \sin^2 x \), use trigonometric identities to simplify the integrand before performing the integration.

Answer 1

The Correct Option is A

We are given the definite integral \( \int_0^{\pi} \sin^2 x \, dx \), and we need to evaluate it. Step 1: Use a standard trigonometric identity We can use the identity for \( \sin^2 x \): \[ \sin^2 x = \frac{1 - \cos(2x)}{2} \] So, the integral becomes: \[ \int_0^{\pi} \sin^2 x \, dx = \int_0^{\pi} \frac{1 - \cos(2x)}{2} \, dx \] Step 2: Break the integral into two parts \[ = \frac{1}{2} \int_0^{\pi} 1 \, dx - \frac{1}{2} \int_0^{\pi} \cos(2x) \, dx \] Step 3: Evaluate the integrals - The first integral is straightforward: \[ \int_0^{\pi} 1 \, dx = x \Big|_0^{\pi} = \pi \] - The second integral involves the cosine function: \[ \int_0^{\pi} \cos(2x) \, dx = \frac{\sin(2x)}{2} \Big|_0^{\pi} = \frac{\sin(2\pi)}{2} - \frac{\sin(0)}{2} = 0 \] Step 4: Combine the results Now, substituting the results back into the expression: \[ \int_0^{\pi} \sin^2 x \, dx = \frac{1}{2} \times \pi - \frac{1}{2} \times 0 = \frac{\pi}{2} \] Thus, the value of the integral is \( \frac{\pi}{2} \). Answer: The value of the integral is \( \frac{\pi}{2} \), so the correct answer is option (1).