Integration of sec^x(x)/√tanx?

1 Answer
May 19, 2018

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))