Evaluate the following indefienite integral ?

Question

#int##(dx)/(x^((3)/(2)) + sqrtx#

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Answer 1 · 2018-04-12T12:44:23

# 2tan^(-1) sqrt x + C#

Start with this sub to clean it up a bit:

#y = sqrt x#, ie # x = y^2#.

Your integral becomes:

#int (d(y^2))/(y^3 + y) = 2 int (dy)/(y^2 + 1)#

This is a well-known problem/ pattern. Use the subsitiution #y = tan z #:

#= 2 int (d (tan z))/(tan^2 z + 1) = 2 int dz = 2 z + C#

So your result is:

#2tan^(-1) y + C = 2tan^(-1) sqrt x + C#