integrals
evaluate the definite integral from 1 to 9 of ((x-1) / (sqrt x))
dx ? Would this one be a u sub?
evaluate the integral from 0 to 10 of abs value (x-5) dx? I think this one would be split but not sure how or why?
∫[1,9] (x-1)/√x dx
let u^2 = x
2u du = dx
Then you have
∫[1,3] (u^2-1)/u * 2u du = 2 ∫[1,3] u^2-1 du = 2 (1/3 u^3 - u)[1,3] = 40/3
But you don't really need a u substitution, since
(x-1)/√x is just x^(1/2) - x^(-1/2)
so the integral is 2/3 x^(3/2) - 2x^(1/2)
To evaluate the definite integral ∫[(x-1)/(√x)]dx from 1 to 9, we can indeed use a u-substitution method. Here's how:
1. Let u = √x. Then, differentiating both sides with respect to x gives du/dx = (1/2) * (1/√x).
2. Rearrange the equation above to solve for dx, so dx = 2√x * du.
3. Substitute the value of x in terms of u: x = u^2.
4. Now rewrite the given integral in terms of u:
∫[(x-1)/(√x)]dx = ∫[(u^2-1)/u] * (2√x * du) = 2∫[(u^2-1)du].
5. Integrate the expression with respect to u:
= 2[∫u^2 du - ∫1 du].
6. Evaluate each integral:
= 2[(1/3)u^3 - u] + C,
where C is the constant of integration.
7. Substitute back u = √x:
= 2[(1/3)(√x)^3 - √x] + C.
8. Evaluate the definite integral by subtracting the value at the lower limit from the value at the upper limit:
= 2[(1/3)(√9)^3 - √9] - 2[(1/3)(√1)^3 - √1].
9. Simplify the expression further:
= 2[(1/3)(3)^3 - 3] - 2[(1/3)(1)^3 - 1].
10. Compute the final result:
= 2[(1/3)(27) - 3] - 2[(1/3)(1) - 1] = 2[9 - 3] - 2[1 - 1] = 12.
Therefore, the value of the definite integral from 1 to 9 of [(x-1)/(√x)]dx is 12.
Now, let's consider the integral of the absolute value of x-5 from 0 to 10: ∫|x-5| dx.
To evaluate this integral, we do indeed need to split it into two separate integrals due to the absolute value function.
1. First, determine the interval where the expression x - 5 is non-negative and the interval where it is negative.
2. For x ≥ 5 (non-negative), the absolute value of x-5 simplifies to x-5. So the integral is:
∫(x-5) dx from 5 to 10.
3. Integrate the expression with respect to x:
= [(1/2)x^2 - 5x] evaluated from 5 to 10.
4. Substitute the upper and lower limits:
= [(1/2)(10^2) - 5(10)] - [(1/2)(5^2) - 5(5)].
5. Simplify the expression:
= [50 - 50] - [12.5 - 25].
6. Calculate the final result:
= 0 - (-12.5) = 12.5.
7. Next, for x < 5 (negative), the absolute value of x-5 simplifies to -(x-5). So, the integral is:
∫-(x-5) dx from 0 to 5.
8. Integrate the expression with respect to x:
= -[(1/2)x^2 - 5x] evaluated from 0 to 5.
9. Substitute the upper and lower limits:
= -[(1/2)(5^2) - 5(5)] - [0 - 0].
10. Simplify the expression:
= -[12.5 - 25] - [0 - 0].
11. Calculate the final result:
= -(-12.5) - 0 = 12.5.
12. Finally, add the values of the two integrals from steps 6 and 11:
12.5 + 12.5 = 25.
Thus, the value of the definite integral from 0 to 10 of |x-5| dx is 25.