Question

The value of 

\[ \int_{0}^{\pi} |\sin^3 x| \, dx \]

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: Simplify the Integral 

On \([0,\pi]\), \( \sin x \ge 0 \), so \[ |\sin^3 x| = \sin^3 x \]

Step 2: Integration Technique

\[ \int \sin^3 x\,dx = \int (1 - \cos^2 x)\sin x\,dx \]

Let \( u = \cos x \), so \( du = -\sin x\,dx \)

Step 3: Evaluate

\[ - \int_{1}^{-1} (1 - u^2)\,du = \int_{-1}^{1} (1 - u^2)\,du \]

\[ = \left[ u - \frac{u^3}{3} \right]_{-1}^{1} \]

\[ = \left(1 - \frac{1}{3}\right) - \left(-1 + \frac{1}{3}\right) = \frac{2}{3} + \frac{2}{3} = \frac{4}{3} \]

Final Answer: (C)