# Thread: ntegrate, but first make a substituition??? HELP!!!

1. ## ntegrate, but first make a substituition??? HELP!!!

my book is asking me to "first make a substitution and then use integration by parts to evaluate the target" the integral i am working on is

x^5 * cos(x^3)

2. I think they want you to to substitute y = x^3. Try it and see what happens.

3. I don't understand how i am supposed to substitute it in, and what is y?

4. Originally Posted by pcgamer03
I don't understand how i am supposed to substitute it in, and what is y?
y is the variable you're changing the problem to... question: do you know what "integration by substitution" means?

5. i do know what it means, i just can't figure you how you would to that and then integrate by parts. If i substitue u = x^3. i get du = 3x^2, or 1/(3x^2)du = dx. so now i need to integrate x^5 * cos(u) 1/(3x^2)du ???

6. i do know what it means, i just can't figure you how you would to that and then integrate by parts. If i substitue u = x^3. i get du = 3x^2, or 1/(3x^2)du = dx. so now i need to integrate x^5 * cos(u) 1/(3x^2)du ???
Excellent, you have the first step. Now you need to simplify (the numerator and denominator have a common factor) and then I think you will see the next step to complete the substitution part.

Once you have gotten rid of all of the xs, you can integrate by parts normally.

7. Originally Posted by pcgamer03
my book is asking me to "first make a substitution and then use integration by parts to evaluate the target" the integral i am working on is

x^5 * cos(x^3)

$\displaystyle \int x^5 \cos \left( x^3 \right)~dx = \int x^3 \cdot x^2 \cos \left( x^3 \right)~dx$

we proceed by substitution

Let $\displaystyle u = x^3$

$\displaystyle \Rightarrow du = 3x^2~dx$

$\displaystyle \Rightarrow \frac 13 du = x^2 ~dx$

So our integral becomes:

$\displaystyle \frac 13 \int u \cos u~du$

now proceed with integration by parts