Question

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

Answer 1

$\int co{s}^{5}xdx=\int co{s}^{4}x.cosxdx$

$=\int (co{s}^{2}x{)}^{2}cosxdx$

$=\int (1-si{n}^{2}x{)}^{2}cosxdx$

Let $sinx=t\Rightarrow cosxdx=dt$

$=\int (1-t^2{)}^{2}dt$

$=\int (1+t^4-2t^2)dt$

$=\int dt+\int t^{4}dt-2\int t^{2}dt$

$=t+\frac{t^5}{5}-\frac{2t^3}{3}+C$

$=\frac{t^5}{5}-\frac{2t^3}{3}+t+c$

$=\frac{si{n}^{5}x}{5}-\frac{2si{n}^{3}x}{3}+sinx+c$

where c is the constant of integration