Consider the integral 8x^2 (x^3+1) dx . In the following, we will evaluate the integral using two methods.
A:
First, rewrite the integral by multiplying out the integrand:
=?
B:
Then evaluate the resulting integral term-by-term:
=?
C:
Next, rewrite the integral using the substitution w=x^3+1
=?
D:
Evaluate this integral (and back-substitute for ) to find the value of the original integral:
=?
E:
How are your expressions from parts (A) and (B) different? What is the difference between the two? (Ignore the constant of integration.)

Any help with this would be much appreciated!

Originally Posted by THKelly

Consider the integral 8x^2 (x^3+1) dx . In the following, we will evaluate the integral using two methods.
A:
First, rewrite the integral by multiplying out the integrand:
=?
B:
Then evaluate the resulting integral term-by-term:
=?
C:
Next, rewrite the integral using the substitution w=x^3+1
=?
D:
Evaluate this integral (and back-substitute for ) to find the value of the original integral:
=?
E:
How are your expressions from parts (A) and (B) different? What is the difference between the two? (Ignore the constant of integration.)

Any help with this would be much appreciated!
A/B: $\int (8x^5+8x^2) dx$ can you integrate this term by term?

C: if $w=x^3+1$ then $dw=3x^2dx$. Can you take it from here

I think you mean to compare the answers from (B) and (D), which should differ only by a constant. After all, there is only one integral of that function, right?

Are you having trouble with the calculations in (A) through (D)? If so, tell us where you're having trouble.

- Hollywood