Find the integral \(\int sin\frac{5}{2}x dx\)

$\frac{-2}{5}\cos {\frac{5}{2}}x+c$

The correct answer is Option (4) → $\frac{-2}{5}\cos {\frac{5}{2}}x+c$

Use substitution:

$t = \frac{5x}{2} \Rightarrow dt = \frac{5}{2}dx \Rightarrow dx = \frac{2}{5}dt$

$\int \sin \left( \frac{5x}{2} \right) dx = \int \sin t \cdot \frac{2}{5} dt$

$= \frac{2}{5} \int \sin t \, dt = \frac{2}{5} (-\cos t)$

$= -\frac{2}{5} \cos \left( \frac{5x}{2} \right) + C$