Question

\[\int \cos^5 x \text{ dx }\]

Answer 1

∫ cos⁵ x dx
= ∫ cos⁴ x . cos x dx
= ∫ (1 – sin² x)² cos x dx

Let sin x = t
⇒ cos x dx = dt
Now, ∫ (1 – sin² x)² cos x dx

= ​​∫ (1 – t²)² . dt
= ∫ (1 + t⁴ – 2t²) dt
= ∫ dt + ​∫ t⁴ dt – 2 ​∫t² dt

\[= t + \frac{t^5}{5} - \frac{2 t^3}{3} + C\]
\[ = \sin x + \frac{\sin^5 x}{5} - \frac{2}{3} \sin^3 x + C\]