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Question
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Solution
Let I = `int(1)/(sqrt(x) + sqrt(x^3)).dx`
= `int(1)/(x^(1/2)+ x^(3/2)).dx`
Put x = t2
∴ dx = 2t dt
Also `x^(1/2) = (t^2)^(1/2)` = t
and
`x^(3/2) = (t^2)^(3/2)` = t3
∴ I = `int (2tdt)/(t + t^3)`
= `2int "tdt"/(t(1 + t^2)`
= `2int (1)/(1 + t^2)dt`
= 2tan–1 t+ c
= `2tan^-1(sqrt(x)) + c`.
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