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Question
`int x^7/(1 + x^4)^2 "d"x`
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Solution
Let I = `int x^7/(1 + x^4)^2 "d"x`
= `int (x^4*x^3)/(1 + x^4)^2 "d"x`
Put 1 + x4 = t
∴ 4x3 dx = dt
∴ x3 dx = `1/4 "dt"`
∴ I = `1/4 int (("t" - 1))/"t"^2 "dt"`
= `1/4(int 1/"t" "dt" -int 1/"t"^2 "dt")`
= `1/4[log|"t"| - "t"^-1/(-1)] + "c"`
∴ `1/4[log|1 + x^4| + 1/(1 + x^4)] + "c"`
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