Integrate from 1 to 5 of (3x-5)^5 dx = Integrate from a to b of f(u) du
where (I have solved this part)
u = 3x-5
du = 3
a = 0
b = 12
The original value of the integral is 165888 via calculator
here's my last question, and it has to be in terms of u:
f(u) = ? (Again, in terms of u)
I originally thought it was 5(u)^4, then tried u^5, but to no avail. What am I doing wrong?
Nevermind, found the answer. It is (1/3)(u^5).
u = 3x-5, so
du = 3 dx, not just 3
∫[1,5] (3x-5)^5 dx = 55552
Now, subbing in for u, the limits of integration are not 0-12. Rather, we have
1/3 ∫[-2,10] u^5 du = 55552
Where did you get 165888?
To find the function f(u), we need to express the integrand in terms of u. Let's start by substituting u = 3x - 5 back into the integral expression and simplifying it:
∫[1 to 5] (3x - 5)^5 dx = ∫[a to b] f(u) du
Substituting u = 3x - 5:
∫[1 to 5] u^5 dx = ∫[a to b] f(u) du
To express the integrand in terms of u, we need to find dx in terms of du. We can differentiate u = 3x - 5 with respect to x:
du/dx = 3
dx = du/3
Now we can rewrite the integral:
∫[1 to 5] u^5 (du/3) = ∫[a to b] f(u) du
Simplifying the integration limits based on the previous values:
∫[0 to 12] u^5/3 du = ∫[a to b] f(u) du
To solve the integral, we can pull the constant 1/3 outside the integral:
(1/3) ∫[0 to 12] u^5 du = ∫[a to b] f(u) du
Integrating u^5 with respect to u:
(1/3) * [u^6/6] from 0 to 12 = ∫[a to b] f(u) du
Simplifying:
(1/3) * [(12^6/6) - (0^6/6)] = ∫[a to b] f(u) du
Calculating:
(1/3) * [2,985,984 - 0] = ∫[a to b] f(u) du
995,328 = ∫[a to b] f(u) du
Therefore, the constant 995,328 represents the value of the integral when evaluated from a to b.
Hence, f(u) = 995,328.