Question

The value of \(\int_{0}^{\frac{\pi}{2}} \sin^5 x \, dx\) is:

• A. 4/5
• B. 8/15
• C. 4/15
• D. 2/5

Hint:

Exam Tip:
For \(\int_0^{\pi/2} \sin^n x \, dx\):

• If \(n\) is odd, the result is rational.
• If \(n\) is even, the result involves \(\pi\).
• Use the Wallis formula or reduction formula.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to evaluate a definite integral of a power of \(\sin x\) from 0 to \(\pi/2\). This is a standard Wallis integral.

Step 2: Key Formula or Approach:
For \(n\) odd: \[ \int_0^{\frac{\pi}{2}} \sin^n x \, dx = \frac{(n-1)(n-3) \cdots 2}{n(n-2) \cdots 1} \] For \(n = 5\): \[ \int_0^{\frac{\pi}{2}} \sin^5 x \, dx = \frac{4 \cdot 2}{5 \cdot 3 \cdot 1} = \frac{8}{15} \]

Step 3: Detailed Explanation:
We can also use the reduction formula: \[ I_n = \int_0^{\frac{\pi}{2}} \sin^n x \, dx = \frac{n-1}{n} I_{n-2} \] With \(I_1 = 1\) and \(I_0 = \frac{\pi}{2}\).
For \(n = 5\): \[ I_5 = \frac{4}{5} I_3 = \frac{4}{5} \cdot \frac{2}{3} I_1 = \frac{4}{5} \cdot \frac{2}{3} \cdot 1 = \frac{8}{15} \]

Step 4: Final Answer:
Therefore, option (B) is correct.