Question #2d11a

1 Answer
Cesareo R.
Jan 16, 2017

#int((1+root(4)(x))/(x+sqrt(x)))dx =2log(sqrt(x)+1)-4arctan(root(4)(x))+4 root(4)(x)+C#

Explanation:

Making #y = root(4)(x)# with #dy=1/4 x^(-3/4)dx# we have

#dx = 4y^3dy# then

#int((1+root(4)(x))/(x+sqrt(x)))dx equiv4 int((1+y)/(y^4+y^2))y^3dy =#

#=4 int((y(1+y))/(y^2+1))dy = 4int((y-1)/(y^2+1)+1)dy=2log(y^2+1)-4arctan(y)+4y+C#

or substituting back

#int((1+root(4)(x))/(x+sqrt(x)))dx =2log(sqrt(x)+1)-4arctan(root(4)(x))+4 root(4)(x)+C#

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