The integral \[\int\limits_0^\pi {{{\sin }^3}\theta \,{\text{d}}\theta } \] is given by A. \[\frac{1}{2}\] B. \[\frac{2}{3}\] C. \[\frac{4}{3}\] D. \[\frac{8}{3}\]

April 15, 2024 by rawan239

/> \int_0^\pi \sin^3 \theta \, d\theta &= \left[ \frac{1}{4} \sin \theta – \frac{3}{8} \cos \theta + \frac{1}{12} \sin 3\theta \right]_0^\pi \
&= \frac{1}{4} – \frac{3}{8} + \frac{1}{12} – \left( \frac{0}{4} – \frac{3}{8} + \frac{0}{12} \right) \
&= \frac{1}{4} – \frac{3}{8} + \frac{1}{12} – \frac{1}{4} + \frac{3}{8} + \frac{0}{12} \
&= \frac{2}{3}
\end{align}$$

Therefore, the integral $\int_0^\pi \sin^3 \theta \, d\theta$ is given by $\boxed{\frac{2}{3}}$.

The other options are incorrect because they do not give the correct value of the integral.