Question

How do you integrate `int 1/(x^2sqrt(x^2-7))` by trigonometric substitution?

Answer 1

`sqrt(x^2-7)/(7x)+C`

Explanation:

`I=intdx/(x^2sqrt(x^2-7))`

Apply the substitution `x=sqrt7sectheta,dx=sqrt7secthetatanthetad theta`.

`color(white)I=int(sqrt7secthetatanthetad theta)/(7sec^2thetasqrt(7sec^2theta-7))`

`color(white)I=1/7int(tanthetad theta)/(secthetasqrt(sec^2theta-1))`

`color(white)I=1/7int(d theta)/sectheta`

`color(white)I=1/7intcosthetad theta`

`color(white)I=1/7sintheta`

`color(white)I=1/7tantheta/sectheta`

`color(white)I=1/7sqrt(sec^2theta-1)/sectheta`

Using `x=sqrt7sectheta=>sectheta=x/sqrt7`:

`color(white)I=1/7sqrt(x^2/7-1)/(x/sqrt7)`

`color(white)I=1/7(1/sqrt7sqrt(x^2-7))/(x/sqrt7)`

`color(white)I=sqrt(x^2-7)/(7x)+C`