Question

\[\int \sin^4 x \cos^3 x \text{ dx }\]

Answer 1

∫ sin⁴ x cos³ x dx
=​ ∫ sin⁴ x . cos² x cos x dx
= ∫ sin⁴ x . (1 – sin² x ) cos x dx

Let sin x = t
⇒​​ cos x dx = dt
Now, ∫ sin⁴ x . (1 – sin² x ) cos x dx

= ​​∫ t⁴ (1 – t²) dt
= ∫ (t⁴ – t⁶) dt

\[= \frac{t^5}{5} - \frac{t^7}{7} + C\]
\[ = \frac{\sin^5 x}{5} - \frac{\sin^7 x}{7} + C\]