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## Homework Statement

here is the answer

## The Attempt at a Solution

My book doesn't do a good job of explaining the substitution rule. here is their explanation:

using the solution manual and looking at how they got the answer to other questions, i've written down my own method in english that i can understand. the method has worked for two other problems, but it broke down with the above problem.

i still don't understand what integrate with respect to u means.

here's my method:

1. substitute one part of the integral with u, find the derivative of that, z,

2. multiply the whole integral by the reciprocal of z, so that z equals 1

3. find the antiderivative of the remaining integral

4. replace u by g(x) in the result

the derivative of g(x) is

step 1. 4y^3 + 8y

multiply 12 by y^3 + 2y

step 2. 12y^3 + 2y

multiply the reciprocal of 1 with the result of step 2

step 3. (12y^3 + 24y)/(4y^3 + 8y)

simplify step 3

step 4. 3 + 3 = 6

we now have

step 5. 6u^2

take the antiderivative of 6u^2

step 6. (6u^3)/3

simplify

step 7 2u^3

plug in g(x) into u

step 8. 2(y^4 + 4y^2 + 1)^3

The book says that the answer is

(y^4 + 4y^2 + 1)^3