∫1a2-x2dx=12a× ______. - Mathematics and Statistics

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`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.

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Solution

`int 1/(a^2 - x^2) dx = 1/(2a) xx` `bb(underline(log|(a + x)/(a - x)| + c)`.

Explanation:

`int1/(a^2 - x^2)dx = int1/((a - x)(a + x))dx`

= `1/(2a)int((a - x) + (a + x))/((a - x)(a + x))dx`

= `1/(2a)int(1/(a + x) + 1/(a - x))dx`

= `1/(2a)[int1/(a + x)dx + int1/(a - x)dx]`

= `1/(2a)[(log|a + x| + (log|a - x|))/-1] + c = 1/(2a)[log|a + x| - log|a - x|] + c`

= `1/(2a)log |(a + x)/(a - x)| + c`

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