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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