Find the value of the definite integral:
int= integral sign
int((x + 2)/(x^(3/2)),x= 16..25)dx
I have no idea how to start this... would I get rid of the fraction?
you do it by converting it to int[x/x^(3/2)+2/x^(3/2)]= int[1/x^(1/2)+2/x^(3/2)]=
int[x^(-1/2)+2*x^(-3/2)]=
[2x^(1/2)-4*x^(-1/2)]x=16...25=
10-4/5-8+1=
11/5
Oops, I should have been more specific. I need to solve it using sums, not antiderivates.
To find the value of the definite integral ∫((x + 2)/(x^(3/2))) dx from 16 to 25, you can start by simplifying the integrand, if possible. In this case, you have a rational function with a square root in the denominator.
To simplify, you can rewrite x^(3/2) as √x^3.
So, the integrand becomes ((x + 2)/√x^3).
Now, let's focus on solving the integral by applying the properties and rules of integrals.
One useful property is that the integral of a sum is the sum of the integrals. So, you can split the integrand into two separate integrals:
∫((x + 2)/√x^3) dx = ∫(x/√x^3) dx + ∫(2/√x^3) dx
Now, let's solve these integrals one by one:
First, let's simplify the first integral: ∫(x/√x^3) dx
To simplify this, you can move x^(-1/2) from the denominator to the numerator, which gives:
∫(x/√x^3) dx = ∫(x^(1/2)/x^((3/2))) dx = ∫ (x^(-1/2)) dx
Now, you can apply the power rule for integrals, which states that ∫x^n dx = (x^(n+1))/(n+1), where n is not equal to -1.
Using the power rule, the integral becomes:
∫ (x^(-1/2)) dx = (x^((1/2)+1))/((1/2)+1) = 2√x
Now, let's move on to the second integral: ∫(2/√x^3) dx
Here, the constant 2 can be taken out of the integral:
∫(2/√x^3) dx = 2 ∫(1/√x^3) dx
To simplify this, you can rewrite x^3 as √(x^6):
2 ∫(1/√x^3) dx = 2 ∫(1/√(x^6)) dx
Using the power rule, the integral can be further simplified as:
2 ∫(1/√(x^6)) dx = 2 ∫(x^(-3/2)) dx = 2 * (-1/(-3/2+1)) * x^(-3/2+1)
Simplifying this expression, we get:
2 * (2/3) * x^(-1/2) = (4/3) * x^(-1/2) = (4/3) * (1/√x)
Now, let's evaluate the definite integral from 16 to 25 by substituting the limits:
∫((x + 2)/(x^(3/2))) dx from 16 to 25 = [2√x] from 16 to 25 + [(4/3) * (1/√x)] from 16 to 25
Now, let's plug in the limits:
[2√(25)] - [2√(16)] + [(4/3) * (1/√25)] - [(4/3) * (1/√16)]
Simplifying, we have:
[2*5] - [2*4] + [(4/3)*(1/5)] - [(4/3)*(1/4)]
= 10 - 8 + (4/15) - (1/3)
= 2 + (4/15) - (1/3)
= 30/15 + (4/15) - (5/15)
= (30 + 4 - 5)/15
= 29/15
Therefore, the value of the definite integral ∫((x + 2)/(x^(3/2))) dx from 16 to 25 is 29/15.