Question

How proof this answer is right ? ∫ ((1)/((x^2)(sqrt(1+x^2))dx =

Answer 1

Explanation:

Clear image of the equation

Answer 2

Show that the derivative of what you think is the answer is the integrand in the original problem.

Explanation:

Quick example:
To prove that `inte^(3x) dx = 1/3 e^(3x)+C`, show that the derivative of `1/3 e^(3x)+C` is `e^(3x)`:

`1/3 e^(3x)*3 = e^(3x)`. So the answer is correct.

In your question, show that the derivative of `-sqrt(1+x^2)/x +C` is `1/(x^2sqrt(1+x^2))`

It is clear that we can ignore the `+C` because the derivative of a constant is `0`.

Answer 3

You could do the actual integral and check each step along the way.

`int 1/(x^2sqrt(1+x^2))dx = ?`

Let:
`x = tantheta`
`dx = sec^2thetad theta`
`sqrt(1+x^2) = sqrt(1+tan^2theta) = sectheta`
`x^2 = tan^2theta`

This is a typical trig substitution strategy and is verified in any calculus textbook that teaches this. You can check the `sqrt(1+x^2)` substitution and you will see that it is really `sqrt(1+tan^2theta) = sqrt(sec^2theta) = sectheta`.

Therefore you get:

`=> int 1/(tan^2thetasectheta)sec^2thetad theta`

Using identities `tantheta = sintheta/costheta` and `sectheta = 1/costheta`:
`= int cos^2theta/sin^2theta*1/costhetad theta`

`= int costheta/sin^2thetad theta`

Some u-substitution can be done now as well. Let:
`u = sintheta`
`du = costhetad theta`

`=> int 1/u^2du`

`= -1/u = color(green)(-1/sintheta)`

That is our temporary answer. Finally, going back to the original substitution of `x = tantheta`:
`tantheta = sintheta/costheta = x`

`sintheta = xcostheta`

`sectheta = sqrt(1+x^2) => costheta = 1/sqrt(1+x^2)`

`sintheta = x/sqrt(1+x^2) => 1/sintheta = sqrt(1+x^2)/x`

Therefore the final answer is indeed:

`= -1/sintheta + C = color(blue)(-sqrt(1+x^2)/x + C)`