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Question
Integrate the following with respect to x.
`1/(x^2(x^2 + 1))`
Solution
`1/(x^2(x^2 + 1)) = x/(x^2(x^2 + 1))`
Let x2 = t
Then 2x dx = dt
So `int 1/(x^2(x^2 + 1)) "d"x = 1/2 int (2x "d"x)/(x^2(x^2 + 1))`
= `1/2 int "dt"/("t"("t" + 1))`
= `1/2 int [1/"t" - 1/("t" + 1)] "dt"` ......(Using partial fractions)
= `1/2 [log |"t"| - log |"t" + 1|] + "c"`
= `1/2 log |"t"/("t" + 1)|`
Replacing t by x2 we get = `1/2 log |x^2/(x^2 + 1)| + "c"`
= `log|x| - 1/2 log|x^2 + 1| + "c"`
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