Question

Question #2250e

Answer 1

`1/2"arcsec"^2(x)+C`

Explanation:

`I=int("arcsec"(x))/(xsqrt(x^2-1))dx`

Let `x=sectheta`. This implies that `dx=secthetatanthetad theta`. Also it means that `theta="arcsec"(x)`. Then:

`I=inttheta/(secthetasqrt(sec^2theta-1))(secthetatanthetad theta)`

Note that `sec^2theta-1=tan^2theta`:

`I=inttheta/(secthetatantheta)(secthetatantheta)d theta`

`I=intthetacolor(white).d theta`

`I=1/2theta^2+C`

`I=1/2"arcsec"^2(x)+C`