Evaluate the definite integral:

upper number of the integral is 6 lower number is 2(4x^2+5)/sqrt of x*dx

To evaluate the definite integral, we can follow these steps:

Step 1: Simplify the integrand.
The integrand in this case is (4x^2 + 5) / √x. To simplify it, we need to rewrite the √x term as x^(1/2). So, the integrand becomes (4x^2 + 5) / x^(1/2).

Step 2: Divide the terms in the numerator by the denominator.
Now, we have (4x^2 / x^(1/2)) + (5 / x^(1/2)). Simplifying further, we get 4x^(2 - 1/2) + 5x^(-1/2) = 4x^(3/2) + 5x^(-1/2).

Step 3: Integrate the simplified integrand.
To integrate 4x^(3/2) + 5x^(-1/2), we use the power rule.

∫ (4x^(3/2) + 5x^(-1/2)) dx = (4/((3/2)+1)) * x^((3/2)+1) + (5/((-1/2)+1)) * x^((-1/2)+1) + C

Removing the fractions inside the exponents:

= (8/3) * x^(5/2) + 10 * x^(1/2) + C

Step 4: Evaluate the definite integral using the Fundamental Theorem of Calculus.
Since we have the limits of integration as 6 (upper limit) and 2 (lower limit), we can now evaluate the definite integral.

∫[2,6] (4x^2 + 5) / √x dx = [(8/3) * x^(5/2) + 10 * x^(1/2)] [2,6]

Substituting the upper limit (6) into the expression:

= [(8/3) * (6^(5/2)) + 10 * (6^(1/2))]

Substituting the lower limit (2) into the expression:

- [(8/3) * (2^(5/2)) + 10 * (2^(1/2))]

Now, perform the calculations to find the final numerical value.