Question

The integral $\int se{c}^{2/3}xcose{c}^{4/3}xdx$ is equal to (Hence $C$ is a constant of integration)

• A. $3ta{n}^{-1/3}x+C$
• B. $-\frac{3}{4}ta{n}^{-4/3}x+C$
• C. $-3co{t}^{-1/3}x+c$
• D. $-3ta{n}^{-1/3}x+C$

Answer 1

Answer: (D)

$I=\int \frac{dx}{(sinx{)}^{4/3}.(cosx{)}^{2/3}}$
$I=\int \frac{dx}{(\frac{sinx}{cosx}{)}^{4/3}.co{s}^{2}x}$
$\Rightarrow I=\int \frac{se{c}^{2}x}{(tanx{)}^{4/3}}dx$
put tanx = t $\Rightarrow se{c}^{2}xdx=dt$
$\therefore I=\int \frac{dt}{t^{4/3}}\Rightarrow I=\frac{-3}{t^{1/3}}+c$
$\Rightarrow I=\frac{-3}{(tanx{)}^{1/3}}+c$