Question

Evaluate the following integral:

\[\int_{- \frac{3\pi}{2}}^{- \frac{\pi}{2}} \left\{ \sin^2 \left( 3\pi + x \right) + \left( \pi + x \right)^3 \right\}dx\]

Answer 1

\[\text{Let I} = \int_{- \frac{3\pi}{2}}^{- \frac{\pi}{2}} \left\{ \sin^2 \left( 3\pi + x \right) + \left( \pi + x \right)^3 \right\}dx\]

Put

\[\pi + x = z\]

\[\Rightarrow dx = dz\]

When

\[x \to - \frac{3\pi}{2}, z \to - \frac{\pi}{2}\]

When

\[x \to - \frac{\pi}{2}, z \to \frac{\pi}{2}\]

\[\therefore I = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \left[ \sin^2 \left( 2\pi + z \right) + z^3 \right]dz\]
\[ = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \left( \sin^2 z + z^3 \right)dz ................\left[ \sin\left( 2\pi + \theta \right) = \sin\theta \right]\]
\[ = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \frac{1 - \cos2z}{2}dz + \int_{- \frac{\pi}{2}}^\frac{\pi}{2} z^3 dz\]

\[= \frac{1}{2} \int_{- \frac{\pi}{2}}^\frac{\pi}{2} dz - \frac{1}{2} \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \cos2zdz + \int_{- \frac{\pi}{2}}^\frac{\pi}{2} z^3 dz\]
\[ = \frac{1}{2} \times z_{- \frac{\pi}{2}}^\frac{\pi}{2} - \left.\frac{1}{2} \times \frac{\sin2z}{2}\right|_{- \frac{\pi}{2}}^\frac{\pi}{2} +\left. \frac{z^4}{4}\right|_{- \frac{\pi}{2}}^\frac{\pi}{2} \]
\[ = \frac{1}{2}\left[ \frac{\pi}{2} - \left( - \frac{\pi}{2} \right) \right] - \frac{1}{4}\left[\sin\pi - \sin\left( - \pi \right) \right] + \frac{1}{4}\left( \frac{\pi^4}{16} - \frac{\pi^4}{16} \right)\]

\[= \frac{1}{2} \times \pi - \frac{1}{4}\left( 0 + 0 \right) + \frac{1}{4} \times 0\]
\[ = \frac{\pi}{2}\]