Question

∫[(e^x)/(e^(2x)+3e^x+2),]

Answer 1

lokisan
u are you replying?

Answer 2

The denominator can be factored as\[(e^x+1)(e^x+2)\]If you then set \[u=e^x \rightarrow du=e^x dx\]you'll end up with\[I=\int\limits_{}{}\frac{e^x}{e^{2x}+3e^x+2}dx=\int\limits_{}{}\frac{du}{(u+1)(u+2)}\]You can use partial fraction decomposition on this last expression:\[\frac{1}{(u+1)(u+2)}=\frac{A}{u+1}+\frac{B}{u+2}\]to find\[A=1, B=-1\]Your integral is then\[I=\int\limits_{}{}\frac{1}{u+1}-\frac{1}{u+2}du = \log (u+1)-\log(u+2)+c\]Substituting back,\[I=\log \frac{e^x+1}{e^x+2}+c\]

Answer 3

is this the answer?

Answer 4

Yes

Answer 5

thank you. this looks so similar to what we are doing

Answer 6

np - I'd appreciate another fan :P

Answer 7

Whenever you see a polynomial in the denominator, try to factor it. Then try to see if you can use partial fraction decomposition.

Answer 8

how do you become a fan? you are a lifesaver. My evil calculas teacher gave us a take home test even i missed the whole week. thank you and can you help with other problems?

Answer 9

there should be a link next to my name - a blue line saying, "Become a fan". If it's not there, maybe try refreshing you page.

Answer 10

dichalao, what the hell's up with your professor giving you that question? It's an algebraic pain.

Answer 11

she's evil

Answer 12

How many more questions do you have?

Answer 13

I have 10 out of which i have done like 3 . 3 of them are set up only using fraction decomposition. and 2 are solving using decomposition

Answer 14

You may want to spread those questions around (i.e. keep making new posts) so others can help too.

Answer 15

do you know how can i type the equation easily like you did in the reply?

Answer 16

There's a button below, "Equation". You can enter things in there.

Answer 17

sorry but can you show the decomposition part?

Answer 18

Loki do u have time to help me out on that problem?

Answer 19

no in the first answer. you said a=1 b=-1

Answer 20

Yes dichalao...just let me finish here and I'll go back.

Answer 21

In partial fraction decomposition, you set it out like that when you have linear factors in the bottom. You then multiply both sides by (u+1)(u+2) and you'll end up with 1 on the left and what you see on the right. You need to find A and B that will make the statement true; that is, by comparing coefficients on the left and right.

Answer 22

thanks