Question

int (e^2x/e^2x 3e^x +2)dx

Answer 1

I assume that is:
\[\int\limits \frac{e^{2x}}{e^{2x}+3e^{x}+2} dx \] ?

Answer 2

yes

Answer 3

Do you know how to write the bottom as a sum/difference of squares?

Answer 4

I've done the U- sub to get \[\int\limits_{}^{}\frac{ 1 }{ u^2 +3u+2}du\]

Answer 5

wait, nvm Im wrong.

Answer 6

hint: \[e^{2x}+3e^{x}+(\frac{3}{2})^2=(e^x+\frac{3}{2})^2 \]

Answer 7

if you want to write e^x as u
then you need to write du as e^x dx

Answer 8

\[\int\limits \frac{e^x e^x}{e^{2x}+3e^{x}+2} dx\]
Replace e^x dx with du
then all other e^x parts get to be u

Answer 9

\[\int\limits_{}^{} \frac{u }{u^2+3u+2} du\]

Answer 10

so I can say the hint now is:
\[u^2+3u+(\frac{3}{2})^2=(u+\frac{3}{2})^2\]

Answer 11

OR! Do you prefer to partial fraction way?

Answer 12

I was going about trig substitution way
We can do partial fractions if you wish since the bottom is easily factored as (u+2)(u+1)

Answer 13

Partial fraction plz

Answer 14

I give you example of how to write the decomposition of this fraction:
\[\frac{u}{u^2+5u+6}=\frac{u}{(u+3)(u+2)}=\frac{A}{u+2}+\frac{B}{u+3} \]
We can combine those fractions to find what the constant A and B are.
\[\frac{u}{(u+3)(u+2)}=\frac{A(u+3)+B(u+2)}{(u+2)(u+3)}\]
We want to find A and B so that these fractions are equal.
The denominators are equal.
Let's find when the numerators are equation.
u=A(u+3)+B(u+2)
u=Au+3A+Bu+2B
<combine like terms
u=(Au+Bu)+3A+2B
u=(A+B)u + (3A+2B)
^
|
no constants on this side
we say the constant is 0 on this side so we have the equation 3A+2B=0
now there is 1u on this side and there is (A+B) u's on the side so A+B=1
Solve these linear equations to find what A and B are.
I like elimination. Let's multiply the bottom equation by -2 giving us:
3A+2B=0
-2A-2B=-2
--------------- <--add the equations
1A+0B=-2
1A=-2
A=-2
So now we can find B by pluggin' into either equation
equation A+B=1 or 3A+2B=0

So if A+B=1 and A=-2, then -2+B=1 which implies B=3.

So we can write
\[\frac{u}{u^2+5u+6} \text{ as } \frac{-2}{u+2}+\frac{3}{u+3}\]
Now this was just an example of how to decompose a fraction.

Answer 15

ty

Answer 16

@myininaya Just to let you know, this was some sort of help but not very much. I know how by parts is done. With this i can not compare my answer to anything and it took a while just to figure this out. This is of very little benefit to me and i have to ask the question again just to get the help I need. Im not trying to be rude but instead of trying to teach perhaps you should just answer the question. That would have been more helpful to me.

Answer 17

Let me know what part you are having trouble with and I will explain it more. I can't help you don't ask any questions on what you are having trouble understanding in the above example.
Like do you need help factoring the bottom?
You will get two linear factors. You write A/(one of the linear factors)+B/(over the other linear factor).