SOLUTION: Hi all, I was hoping to get some help with the following 2 intergrals. I have to evaluate each one using the given substitution in each. a) ∫( (x^2x) / ((3 + x^3)^2) ) Sub

Question 211741: Hi all, I was hoping to get some help with the following 2 intergrals.
I have to evaluate each one using the given substitution in each.
a) ∫( (x^2x) / ((3 + x^3)^2) ) Substitution, u = x^3
b) ∫ \( x^7 cos x^8dx) Substitution u = x^8
Help with steps and notes on how to solve would be gret.
Thanks, -Nick.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
a) ∫( (x^2x) / ((3 + x^3)^2) ) Substitution, u = x^3

I assume this is supposed to be:
a) ∫ Substitution, u = x^3
Actually a better substitution would be: but what you were given will still work. We start by finding the derivative of both sides of :

Multiplying both sides of this by dx we get:

Looking at the original integral we can see the but not the 3. So we can "move" the 3 by multiplying both sides by (or divide both sides by 3):

Now we can substitute u for and for giving:
a) ∫
This can be rewritten as:
a) ∫
If the integral above is not obvious to you, then use a second substitution: v = 3+u. Either way we should get:

b) ∫ \( x^7 cos x^8dx) Substitution u = x^8
I assume the "\" is a typo and that the integral is:
b) ∫
Again we start by finding the derivative of :

Multiplying both sides by dx:

Using the Commutative Property on the original integral we can see the right side of the above equation in the integral:
∫
So we can now substitute u for and du for :
∫
This is a pretty simple integral to find:

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SOLUTION: Hi all, I was hoping to get some help with the following 2 intergrals. I have to evaluate each one using the given substitution in each. a) &#8747;( (x^2x) / ((3 + x^3)^2) ) Sub

SOLUTION: Hi all, I was hoping to get some help with the following 2 intergrals. I have to evaluate each one using the given substitution in each. a) ∫( (x^2x) / ((3 + x^3)^2) ) Sub