Integral of a rational function int(sqrt[3](x)+1)/(sqrt[4](x^3)(sqrt x-sqrt[6](x))dx)

Mark Rosales

Mark Rosales

Answered question

2022-11-12

Integral of a rational function
x 3 + 1 x 3 4 ( x x 6 ) d x

Answer & Explanation

Tasinazzokbc

Tasinazzokbc

Beginner2022-11-13Added 17 answers

Actually, this is beautiful. When you substitute x = t 12 , you get
12 d t t 4 1 t 4 + 1
Use partial fractions to show that
t 4 1 t 4 + 1 = 1 + 1 / 2 t + 1 1 / 2 t 1 + 1 1 + t 2
Then the integral is
12 t + 6 log ( t + 1 t 1 ) + 12 arctan t + C
where C is an integration constant. Substitute back t = x 1 / 12 and you are done.

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