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Question
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
Solution
Let I = `int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
= `int "e"^(log^((x^3)))/(x^4 + 1) "d"x`
= `int x^3/(x^4 + 1) "d"x`
Put x4 + 1 = t
Differentiating w.r.t. x, we get
4x3dx = dt
∴ x3dx = `1/4 "dt"`
∴ I = `1/4 int "dt"/"t"`
= `1/4 log|"t"| + "c"`
∴ I = `1/4 log|x^4 + 1| + "c"`
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