Question

How do you integrate `int sqrt(1+x^2)/xdx` using trigonometric substitution?

Answer 1

`I = sqrt(x^2+ 1) - ln|(sqrt(x^2 + 1) + 1)/x| + C`

Explanation:

We let `x = tantheta`. Then `dx =sec^2theta d theta`.

`I = int sqrt(1 + tan^2theta)/tantheta * sec^2theta d theta`

`I = int sqrt(sec^2theta)/tantheta * sec^2theta d theta`

`I = int sec^3theta/tantheta d theta`

`I = int (1/cos^3theta)/(sintheta/costheta) d theta`

`I = int cscthetasec^2theta`

`I = int csctheta(1 + tan^2theta)d theta`

`I = int csctheta + cscthetatan^2theta d theta`

`I = int csctheta + secthetacostheta d theta`

Now these are two known integrals.

`I = sectheta - ln|csctheta + cottheta| + C`

IF `x/1 = tantheta`, then `sectheta = sqrt(x^2 + 1)` and `csctheta = sqrt(x^2 + 1)/x` and `cottheta = 1/x`.

`I = sqrt(x^2 + 1) - ln|sqrt(x^2 + 1)/x + 1/x| + C`

`I = sqrt(x^2+ 1) - ln|(sqrt(x^2 + 1) + 1)/x| + C`

Hopefully this helps!