Evaluate the definite integral

The S thingy has 1 at the bottom and 9 at the top. 4x^2+5 divided by the sqrt of x.

The S thingy is called the integral sign.

The number at the bottom (1) is the lower limit of a definite integral, and the top number (9) is the upper limit of integration.

The expression to be evaluated probably looks similar to this:

I = ∫19 (4*x^2+5)/sqrt(x) dx

If you use the substitution
u=sqrt(x), then
du=(1/2)*dx/sqrt(x)

Substituting the limits and the variables involving x, we get
I= ∫19 (4*x^2+5)/sqrt(x) dx

= ∫sqrt(x)sqrt(9) (4u^4+5)*2 du
= ∫sqrt(x)sqrt(9) (4u^4+5)*2 du

Continuing the integration and evaluate the integral according to the integration limits, we should obtain 2036/5 as the numerical answer.

Post if you need more details.

Thanks for the help!

In case it confused you, the substituted lower limit should have read sqrt(1).

To evaluate the definite integral, we need to find the antiderivative of the function and then use the Fundamental Theorem of Calculus.

Step 1: Find the antiderivative of the function.
The function we want to integrate is (4x^2 + 5) / sqrt(x). To find the antiderivative of this function, we can split it into two separate terms. The first term, 4x^2 / sqrt(x), can be simplified as 4x^(2-1/2) = 4x^(3/2). The antiderivative of this term is 2/5 * (4/5) * x^(5/2) = (8/25) * x^(5/2). The second term, 5 / sqrt(x), can be simplified as 5 * x^(-1/2). The antiderivative of this term is 5 * (2 * x^(1/2)) = 10x^(1/2).

So, the antiderivative of the function is (8/25) * x^(5/2) + 10x^(1/2).

Step 2: Apply the Fundamental Theorem of Calculus.
According to the Fundamental Theorem of Calculus, the definite integral of a function f(x) from a to b is equal to F(b) - F(a), where F(x) is the antiderivative of f(x).

To evaluate the definite integral of (4x^2 + 5) / sqrt(x) from 1 to 9, we substitute the upper limit (9) into the antiderivative: (8/25) * (9)^(5/2) + 10 * (9)^(1/2).
Next, we substitute the lower limit (1) into the antiderivative: (8/25) * (1)^(5/2) + 10 * (1)^(1/2).

Simplifying these expressions, we have:
Upper limit: (8/25) * (9^(5/2)) + 10 * (9^(1/2))
Lower limit: (8/25) * (1^(5/2)) + 10 * (1^(1/2))

Now, subtract the lower limit from the upper limit:
[(8/25) * (9^(5/2)) + 10 * (9^(1/2))] - [(8/25) * (1^(5/2)) + 10 * (1^(1/2))].

Evaluating this expression will give you the value of the definite integral.