The integral ^2/3x^4/3x,dx is equal to: (Here C is a constant of integration)

Question

The integral $\displaystyle\int\sec^{2/3}x\csc^{4/3}x\,dx$ is equal to: (Here C is a constant of integration)

Accepted Answer

$$I=\int\sec^{2/3}x\csc^{4/3}x\,dx=\int\frac{\sec^2x}{\tan^{4/3}x}dx$$

Put $\tan x=z\Rightarrow\sec^2x\,dx=dz$:
$$I=\int z^{-4/3}dz=\frac{z^{-1/3}}{-1/3}+C=-3(\tan x)^{-1/3}+C$$

Solutions · 1