Evaluate the definite integral:
[4,8] (4x^2+10)/(x^(1/2)) dx
Been trying this for about 1 hour. Very frustrated, thank you.
The integrand is the sum of two terms,
4 x^(3/2) + 10 x^(-1/2)
Integrate them separately and add the results. The indefinite integral is
(8/5) x^(5/2) + 20 x^(1/2)
Evaluate that at x=8 and subtract the value of the same function at x = 4.
Hint:
get rid of the denominator
(4x^2 + 10)/(x^(1/2))
=4x^(3/2) + 10x^(-1/2)
glad we agree so far
To evaluate the definite integral ∫[4,8] (4x^2+10)/(x^(1/2)) dx, we can start by simplifying the integrand as much as possible.
The first step is to rewrite x^(1/2) as √x. Therefore, we have:
∫[4,8] (4x^2+10)/(√x) dx
Next, we can split the integral into two separate integrals using the property of addition:
∫[4,8] (4x^2)/(√x) dx + ∫[4,8] (10)/(√x) dx
We can simplify the first integral by using the exponent rule for dividing terms with the same base. The rule states that x^m / x^n = x^(m-n). Applying this rule, we get:
∫[4,8] (4x^(2-1/2)) dx + ∫[4,8] (10)/(√x) dx
Now, we can simplify further:
∫[4,8] 4x^(3/2) dx + ∫[4,8] (10)/(√x) dx
To find the antiderivative of 4x^(3/2), we add 1 to the exponent and divide the coefficient by the new exponent:
(4/(3/2 + 1)) x^(3/2 + 1) = (8/5) x^(5/2)
So, the first integral becomes:
∫[4,8] (8/5) x^(5/2) dx
To find the antiderivative of (10)/(√x), we can rewrite it as 10x^(-1/2) and then apply the exponent rule:
(10/(1/2 + 1)) x^(-1/2 + 1) = 20x^(1/2)
So, the second integral becomes:
∫[4,8] 20x^(1/2) dx
We can now evaluate each integral separately:
∫[4,8] (8/5) x^(5/2) dx = (8/5) * (2/7) * x^(7/2) | [4,8]
= (16/35) * (8^(7/2) - 4^(7/2))
∫[4,8] 20x^(1/2) dx = (20/(1/2 + 1)) * (2/3) * x^(3/2) | [4,8]
= (40/3) * (8^(3/2) - 4^(3/2))
Finally, we can evaluate the definite integral by subtracting the value of the lower limit from the value of the upper limit:
[(16/35) * (8^(7/2) - 4^(7/2))] - [(40/3) * (8^(3/2) - 4^(3/2))]
Calculating this expression will give you the result.