Let y = integrate from 1-9x to 1 of (u^3)/(1+u^2)
What is y'?
I flipped the integral so it's 1 to 1-9x and had the derivative of -9 via chain rule
so -9 and just plugged in 1-9x to the (u^3)/(1+u^2) so it's -9*((1-9x)^3)/(1+(1-9x)^2). It tells me what's wrong; but what is my error?
Hmm. Looks good to me. Did they want you to expand the polynomials?
6561x^3-2187x^2+243x-9
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81x^2-18x+2
Or, since you flipped the limits of integration, that would change the sign of the answer, so it'd be +9*...
Oh, thanks Steve! it's just the +9 that got me off track.
It seems like you made a mistake when applying the chain rule to find the derivative. Let's go through the correct steps to find y' for the given integral.
First, let's start by rewriting the integral:
y = ∫[1-9x]^1 [(u^3)/(1+u^2)] du
Now, to find y', we need to differentiate with respect to x. To do this, we will use the Fundamental Theorem of Calculus and the chain rule. Let's proceed step by step.
Step 1: Rewrite the integral using u-substitution.
Let u = 1 - 9x. To find the limits of integration in terms of u, we substitute the original limits:
When x = 1, u = 1 - 9(1) = -8
When x = 1 - 9x = 1 - 9(0) = 1
So, the integral becomes:
y = ∫[-8]^1 [(u^3)/(1+u^2)] du
Step 2: Differentiate y with respect to x using the chain rule.
We differentiate both sides of the equation with respect to x:
d/dx[y] = d/dx[∫[-8]^1 [(u^3)/(1+u^2)] du]
Step 3: Apply the chain rule.
Let's denote the integrand as f(u) = (u^3)/(1+u^2). We need to find df/du and du/dx.
df/du = 3u^2(1+u^2)^(-1) - (2u^4)(1+u^2)^(-2)
du/dx = -9
Now we can use the chain rule:
d/dx[∫[-8]^1 [(u^3)/(1+u^2)] du] = df/du * du/dx
Step 4: Evaluate the derivatives at the given limits.
Let's substitute u = -8 and u = 1 into the expressions for df/du and du/dx to find the derivatives at the limits.
df/du = 3(-8)^2(1+(-8)^2)^(-1) - (2(-8)^4)(1+(-8)^2)^(-2)
du/dx = -9
Step 5: Multiply the derivatives by the corresponding limits and simplify.
Now we evaluate df/du and du/dx at the limits and multiply them:
df/du * du/dx evaluated at u = -8:
= [3(-8)^2(1+(-8)^2)^(-1) - (2(-8)^4)(1+(-8)^2)^(-2)] * -9
df/du * du/dx evaluated at u = 1:
= [3(1)^2(1+(1)^2)^(-1) - (2(1)^4)(1+(1)^2)^(-2)] * -9
Step 6: Simplify the expression obtained in Step 5.
After evaluating and simplifying the above expression, you will find the correct y' for the integral.
By careful evaluation and simplification, you can find the correct derivative expression for y'.
I hope this explanation helps you understand the process of finding y' for the given integral.