1. ## Integration

I have the following integral.

$\displaystyle \int cos(x) - 2e^x + 3x^-1$

So far I can work out the first and last terms to be, I did this by simplying splitting this up into 3 terms.

$\displaystyle \int sin(x) - 2e^x + 3ln(x) + C$

What I am unsure about is how to integrate the $\displaystyle -2e^x$

2. Originally Posted by gk99
I have the following integral.

$\displaystyle \int cos(x) - 2e^x + 3x^-1$

So far I can work out the first and last terms to be, I did this by simplying splitting this up into 3 terms.

$\displaystyle \int sin(x) - 2e^x + 3ln(x) + C$

What I am unsure about is how to integrate the $\displaystyle -2e^x$
1. You don't need the integration symbol once you have integrated.

2. You're correct in your solution. Recall the property $\displaystyle \int c f(x)dx=c\int f(x)dx$ where $\displaystyle c\in\mathbb{R}$ and since $\displaystyle \frac{d}{dx}\left[e^x\right]=e^x$, it would make sense that $\displaystyle \int e^xdx=e^x+C$.

Does this clarify things?

3. Originally Posted by gk99
I have the following integral.

$\displaystyle \int cos(x) - 2e^x + 3x^-1$

So far I can work out the first and last terms to be, I did this by simplying splitting this up into 3 terms.

$\displaystyle \int sin(x) - 2e^x + 3ln(x) + C$

What I am unsure about is how to integrate the $\displaystyle -2e^x$
$\displaystyle \int{a\,e^{b\,x}\,dx} = \frac{a}{b}\,e^{b\,x} + C$.