# Thread: Integration: General Power Formula

1. ## Integration: General Power Formula

Hi. Question - integrate the following function:

28-1-19

(int) (4 + ((e^x))^3) e^x dx

The answer is 1/4(4 + (e^x))^4 + C

doing the work myself, du = 3((4 + (e^x))^2) which means it does not fit the general power formula, because e^x is not the derivative of (4 + ((e^x))^3)

nor would it work the other way around, the derivative of e^x is e^x correct??

or can someone explain how it does? Thanks.

2. ## Re: Integration: General Power Formula

Hello, togo!

You are choosing the wrong substitution.

$\displaystyle \int (4+e^x)^3\,e^x\,dx$

We have: .$\displaystyle \int(4 + e^x)^3(e^x\,dx)$

Let $\displaystyle u \,=\,4+e^x \quad\Rightarrow\quad du \,=\,e^x\,dx$

Substitute: .$\displaystyle \int u^3\,du \;=\;\tfrac{1}{4}u^4+C$

Back-substitute: .$\displaystyle \tfrac{1}{4}(4 + e^x)^4 + C$

3. ## Re: Integration: General Power Formula

Originally Posted by togo
Hi. Question - integrate the following function:

28-1-19

(int) (4 + ((e^x))^3) e^x dx

The answer is 1/4(4 + (e^x))^4 + C

doing the work myself, du = 3((4 + (e^x))^2) which means it does not fit the general power formula, because e^x is not the derivative of (4 + ((e^x))^3)

nor would it work the other way around, the derivative of e^x is e^x correct??

or can someone explain how it does? Thanks.
$\displaystyle \int \left (4 + \left ( e^x \right ) ^3 \right ) e^x dx$

Now $\displaystyle \left ( e^x \right )^3 = e^{3x}$ so your integral becomes:
$\displaystyle \int \left (4 + \left ( e^x \right ) ^3 \right ) e^x dx = \int \left (4 + e^{3x} \right ) e^x dx$

Now try the substitution $\displaystyle y = e^x$.

Just a thought: If the integral is supposed to be $\displaystyle \int \left ( 4 + e^x \right )^3 e^x dx$ again do the substitution $\displaystyle y = e^x$.

-Dan

4. ## Re: Integration: General Power Formula

looks like I got the brackets wrong, thanks for extrapolating the answer. I see that I used chain rule instead of general power formula.