What is the integral of x*sqrt (2x^2)?

This problem is a little bit tricky. Since it is not given the
range of admissible values for x, we'll solve the integral admitting that interval of values for
x is the real set number.

Since the domain of values for the
function f(x) = x*sqrt(2x^2) is R, then:

sqrt x^2 = |x| and not
x

We'll solve the integral considering 2
cases:

Case 1:

For x>0, |x| =
x

Int x*sqrt(2x^2)dx = (sqrt2)*Int (x^2)dx = [(sqrt2)*x^3]/2 +
C

Case 2:

For x<0. then |x| =
-x

Int x*sqrt(2x^2)dx = (sqrt2)*Int - (x^2)dx = - [(sqrt2)*x^3]/2 +
C

The indefinite integral of the given function is:
[(sqrt2)*x^3]/2 + C, if x > 0, or - [(sqrt2)*x^3]/2 + C, if x <
0.