Question

$\int {\cos }^{5}xdx$

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
$$\begin{gathered} =t+\frac{t^5}{5}-\frac{2t^3}{3}+C \\ =\sin x+\frac{{\sin }^{5}x}{5}-\frac{2}{3}{\sin }^{3}x+C \end{gathered}$$