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)
please help me out, I have no clue what they mean by this
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)
please help me out, I have no clue what they mean by this
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.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 ???
Once you have gotten rid of all of the xs, you can integrate by parts normally.
$\displaystyle \int x^5 \cos \left( x^3 \right)~dx = \int x^3 \cdot x^2 \cos \left( x^3 \right)~dx$
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