Evaluate the `int (dx)/((x^2)(sqrt(1+x^2)))`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The integral `int 1/(x^2*sqrt(1+x^2)) dx` has to be determined.

Let `x = tan y`

`dx = sec^2y dy `

`dx = 1 + tan^2y dy`

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

=> `int (1+tan^2y)/(tan^2y*sqrt(1+tan^2y)) dy`

=> `int sqrt(1 +tan^2y)/(tan^2y) dy`

=> `int sec y/(tan^2y) dy`

=> `int cos y/(sin^2 y) dy`

=> `-1/sin y`

As `x = tan y`

`x = sin y/cos y`

=> `x*cos y = sin y`

=> `x^2*cos^2y = sin^2y`

=> `x^2*(1 - sin^2y) = sin^2y`

=> `x^2 = sin^2y + x^2*sin^2y`

=> `sin^2y = x^2/(1 + x^2)`

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

`-1/sin y`

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

The integral `int 1/(x^2*sqrt(1+x^2)) dx = -sqrt(1 + x^2)/x + C`

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