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: Simplify integral
In ([0, \pi]), (\sin x) is positive, 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), (du = -\sin x , dx).

Step 3: Evaluate
(- \int_{1}^{-1} (1 - u^2) , du = \int_{-1}^{1} (1 - u^2) , du = [u - \frac{u^3}{3}]_{-1}^{1}).
(= (1 - 1/3) - (-1 + 1/3) = 2/3 + 2/3 = 4/3).
Final Answer: (C)