Question

Evaluate the following.

`int 1/(sqrt(3"x"^2 - 5))` dx

Answer 1

Let I = `int 1/(sqrt(3"x"^2 - 5))` dx

`= 1/sqrt3 int 1/sqrt("x"^2 - 5/3)` dx

`= 1/sqrt3 int 1/(sqrt ("x"^2 - (sqrt5/sqrt3)^2))` dx

`= 1/sqrt3 log |"x" + sqrt("x"^2 - (sqrt5/sqrt3)^2)| + "c"_1`

`= 1/sqrt3 log |"x" + sqrt("x"^2 - 5/3)| + "c"_1`

`= 1/sqrt3 log |(sqrt3"x" + sqrt(3"x"^2 - 5))/sqrt3| + "c"_1`

`= 1/sqrt3 log |sqrt3"x" + sqrt(3"x"^2 - 5)| - 1/sqrt3 log sqrt3 + "c"_1`

∴ I = `1/sqrt3 log |sqrt3"x" + sqrt(3"x"^2 - 5)| + "c"`,
where c = `"c"_1 - 1/sqrt3 log sqrt3`