Question

Integration of sec^x(x)/√tanx?

Answer 1

`int sec^x(x)/sqrt(tan(x))dx` has no representation in terms of elementary functions

`int sec^2(x)/sqrt(tan(x))dx=2sqrt(tan(x))+C`

Explanation:

For `int sec^2(x)/sqrt(tan(x))dx`,

Let `u=sqrt(tan(x))`
`(du)/dx=sec^2(x)/(2sqrt(tan(x)))=(sec^2(x))/(2u)`
`dx=(2u)/(sec^2(x)) du`

Substituting,

`int sec^2(x)/sqrt(tan(x))dx=int sec^2(x)/u*(2u)/sec^2(x)du=int 2du=2u+C=2sqrt(tan(x))+C`

Trivia

Other integrals of the form `int sec^n(x)/sqrt(tan(x))dx` like
`int sec^4(x)/sqrt(tan(x))dx=2/5 sqrt(tan(x)) (sec^2(x) + 4)+C`

`int (sec^6(x))/sqrt(tan(x)) dx = 2/45 sqrt(tan(x)) (5 sec^4(x) + 8 sec^2(x) + 32)+C`

can also be done with similar substitution of `u=sqrt(tan(x))`