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Question
Integrate the following with respect to x.
`("e"^(3logx))/(x^4 + 1)`
Solution
`("e"^(3logx))/(x^4 + 1) = ("e"^(log x^3))/(x^4 + 1)`
`x^3/(x^4 + 1)`
Let x4 + 1 = f(x)
Then 4x3 = f'(x)
So `int ("e"^(3logx))/(x^4 + 1) "d"x = int x^3/(x^4 + 1) "d"x`
= `1/4 int (4x^3)/(x^4 + 1) "d"x`
= `1/4 int ("f'"(x))/"f'(x) "d"x`
= `1/4 log |"f"(x)| + "c"`
= `1/4 log|x^4 + 1| + "c"`
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