Thread: Steps in integration problem

Hi, while working on some integration problems today, I came across this one:


I multiplied it out:


(Edit: it's supposed to say x^2+2x in the first term)

And with some trial and error I got an answer:


When I differentiate this I get back what I started out with.

But trial and error isn't usually the best way to get an answer, so can anyone tell me which steps I need to take here?

Should I be using U-substitution?

if $\displaystyle t=x^2+2x\implies dt=(2x+2)\,dx=2(x+1)\,dx.$

Originally Posted by TwoPlusTwo

Hi, while working on some integration problems today, I came across this one:


I multiplied it out:


(Edit: it's supposed to say x^2+2x in the first term)

And with some trial and error I got an answer:


When I differentiate this I get back what I started out with.

But trial and error isn't usually the best way to get an answer, so can anyone tell me which steps I need to take here?

Should I be using U-substitution?

You can do that, or else you can notice that $\displaystyle x+1$ is half the derivative of $\displaystyle x^2+2x$ , so you can write

$\displaystyle \int (x+1)e^{x^2+2x}dx = \frac{1}{2}\int (2x+2)e^{x^2+2x}dx$ and then use the almost-immediate integral

$\displaystyle \int f'(x)e^{f(x)}dx=e^{f(x)}+C$

Tonio

Makes perfect sense. Thanks guys! I had a feeling I was missing something obvious.