Solve integral:                    `int (sqrt(2+x^2)-sqrt(2-x^2))/sqrt(4-x^4) dx`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The integral `int (sqrt(2+x^2) - sqrt(2 - x^2))/sqrt(4 - x^4) dx` has to be determined.

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

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

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

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

The integral `int (sqrt(2+x^2) - sqrt(2 - x^2))/sqrt(4 - x^4) dx` = `sin^-1(x/sqrt(2)) - sinh^-1(x/sqrt(2)) + C`

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