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))#