Question

The value of \[ \int_{0}^{\pi} \left|\sin^{3}x\right| \, dx \] is:

• A. 0
• B. 3/8
• C. 4/3
• D. [suspicious link removed]

Hint:

Wallis' Formula shortcut for $\int_{0}^{\pi/2} \sin^n x \, dx$: If $n=3$, value is $\frac{2}{3}$. For $\int_{0}^{\pi}$, it is twice that: $4/3$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to evaluate a definite integral of a power of the sine function. In the interval \( [0, \pi] \), \( \sin x \) is positive, so the absolute value can be removed.
Step 3: Detailed Explanation:
\[ I = \int_0^\pi |\sin^3 x| \, dx = \int_0^\pi \sin^3 x \, dx \]
By property \( \int_0^{2a} f(x) \, dx = 2\int_0^a f(x) \, dx \) if \( f(2a-x) = f(x) \):
Since \( \sin^3(\pi - x) = \sin^3 x \):
\[ I = 2 \int_0^{\pi/2} \sin^3 x \, dx \]
Using Wallis formula for \( n=3 \):
\[ \int_0^{\pi/2} \sin^n x \, dx = \frac{n-1}{n} \cdot \frac{n-3}{n-2} \dots \]
For \( n=3 \): \( \frac{3-1}{3} \times 1 = \frac{2}{3} \).
So, \( I = 2 \times \frac{2}{3} = \frac{4}{3} \).
Step 4: Final Answer:
The value of the integral is 4/3.