Question

The value of \(\int_0^\pi\left|\sin ^3 x\right| \mathrm{d} x\) is

• A. 0
• B. \(\frac{3}{8}\)
• C. \(\frac{4}{3}\)
• D. \(\pi\)

Hint:

Sine is non-negative on 0 to pi, so the modulus can be dropped; write sin cubed in terms of cos and integrate.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
On \([0,\pi]\), \(\sin x\geq0\), so \(|\sin^3x|=\sin^3x\).

Step 2: Rewrite
\(\sin^3x=\sin x(1-\cos^2x)\). Let \(u=\cos x\), \(du=-\sin x\,dx\).

Step 3: Integrate
\(\int_0^{\pi}\sin^3x\,dx=\int_{-1}^{1}(1-u^2)\,du=\left[u-\frac{u^3}{3}\right]_{-1}^{1}=\frac{2}{3}+\frac{2}{3}=\frac{4}{3}\).

Step 4: Compare the options
This is option (C).

Final Answer:
The value of the integral is \(\frac{4}{3}\). \[ \boxed{\text{(C) }\dfrac{4}{3}} \]