is the integral of the square root of x*ln(x)
=(2/3 x^(3/2))(lnx-2/3)
?
IF you mean
int [ ln (x) * x^(1/2) dx
I get from integral table
x^(3/2) [ ln(x)/(3/2) - 4/9 ]
So yes, we agree.
To verify whether or not the given integral is equal to (2/3 x^(3/2))(lnx-2/3), we can take the derivative of this expression and see if it matches the original function.
First, let's start by assuming the given integral expression is correct:
∫√(x ln(x)) dx = (2/3 x^(3/2))(ln(x) - 2/3)
Next, we need to differentiate the expression on the right side with respect to x:
(d/dx) [(2/3 x^(3/2))(ln(x) - 2/3)]
To do this, we can use the product rule, which states that if we have two functions multiplied together, the derivative of the product is equal to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.
Let's apply the product rule to differentiate the expression:
(d/dx) [(2/3 x^(3/2))(ln(x) - 2/3)]
= (2/3) [d/dx (x^(3/2))(ln(x) - 2/3)] + (ln(x) - 2/3) [d/dx (2/3 x^(3/2))]
Now, let's differentiate each term separately:
[d/dx (x^(3/2))] = (3/2) x^(1/2)
[d/dx (ln(x) - 2/3)] = 1/x
Substituting these derivatives back into the expression:
(2/3) [(3/2) x^(1/2)](ln(x) - 2/3) + (ln(x) - 2/3) (2/3 x^(3/2))
Simplifying:
(1/2) x^(1/2) (ln(x) - 2/3) + (ln(x) - 2/3) x^(3/2)
Differentiating each term separately:
(1/2) [(d/dx (x^(1/2)))(ln(x) - 2/3) + x^(1/2) (d/dx (ln(x) - 2/3))] + [(d/dx (ln(x) - 2/3))(x^(3/2)) + (ln(x) - 2/3)(d/dx (x^(3/2)))]
(d/dx (x^(1/2))) = (1/2) x^(-1/2)
(d/dx (ln(x) - 2/3)) = 1/x
(d/dx (x^(3/2))) = (3/2) x^(1/2)
Substituting these derivatives back into the expression:
(1/2) [(1/2) x^(-1/2)](ln(x) - 2/3) + x^(1/2) (1/x) + (ln(x) - 2/3) (3/2) x^(1/2)
Simplifying further:
(1/4) x^(-1/2)(ln(x) - 2/3) + (1/2) x^(1/2)/x + (3/2) x^(1/2) (ln(x) - 2/3)
Now, let's simplify the middle term:
(1/2) x^(1/2)/x = (1/2) x^(1/2 - 1) = (1/2) x^(-1/2)
Combining like terms:
(1/4) x^(-1/2)(ln(x) - 2/3) + (1/2) x^(-1/2) + (3/2) x^(1/2) (ln(x) - 2/3)
Notice that the first term (1/4) x^(-1/2)(ln(x) - 2/3) is the same as the derivative of the given integral expression.
Therefore, the integral expression (2/3 x^(3/2))(ln(x) - 2/3) is not equal to the integral of √(x ln(x)).