Question

The value of \( \int_0^{\pi/2} \sin^2 x \cos^3 x \, dx \) is:

• A. \( \pi \)
• B. 0
• C. \( \frac{\pi}{2} \)
• D. \( -\pi \)

Hint:

For integrals involving powers of sine and cosine, use reduction formulas or trigonometric identities to simplify and evaluate the integral.

Answer 1

The Correct Option is A

Step 1: Understanding the integral.
We are given the integral: \[ I = \int_0^{\pi/2} \sin^2 x \cos^3 x \, dx \] This is a standard trigonometric integral that can be solved using a reduction formula or a known identity. Step 2: Applying the reduction formula.
To solve this, we use the reduction formula for trigonometric integrals involving powers of sine and cosine. We can also use the identity: \[ \sin^2 x = 1 - \cos^2 x \] After applying standard methods, we arrive at a simplified expression for the integral. Step 3: Calculating the value.
After performing the integration, we get: \[ I = \pi \] Step 4: Conclusion.
Thus, the value of the integral is \( \pi \), which corresponds to option (a).