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Question
Evaluate the following:
`int x^(1/2)/(1 + x^(3/4)) "d"x` (Hint: Put `sqrt(x)` = z4)
Solution
Let I = `int x^(1/2)/(1 + x^(3/4)) "d"x`
Put x = t4
⇒ dx = 4t3d dt
∴ I = `4int ("t"^2("t"^3))/(1 + "t"^3) "dt"`
= `4int("t"^2 - "t"^2/(1 + "t"^3)) "dt"`
= `4 int "t"^2 "dt" - 4 int "t"^2(1 + "t"^3) "dt"`
= `4 "t"^3/3 - 4/3 int (3"t"^2)/(1 + "t"^3) "dt"`
= `4 "t"^3/3 - 4/3 log |1 + "t"^3| + "C"`
= `4/3 x^(3/4) - 4/3 log |(1 + x^(3/4))| + "C"`
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