Question

Indefinite integral need help pleaseeeeeee

Answer 1

Integrate indefinite integral:

`I=\int(dx)/(e^(2x)+3e^x+2)dx`

Solution:
1. use substitution
`u=e^x`
=>

`du=e^xdx`
=>

`dx=(du)/(e^x)`
=>

`dx=(du)/(u)`
=>

`I=\int(du)/(u(u^2+3u+2))du`

`=\int(du)/(u(u+2)(u+1))du`
2. decompose into partial fractions

`(1)/(u(u+2)(u+1))`

`=(A)/(u)+(B)/(u+2)+(C)/(u+1)`
where A=1/2, B=1/2, C=-1

`=(1)/(2u)+(1)/(2(u+2))-(1)/(u+1)`
3. Substitute partial fractions and continue

`I=\int(du)/(2u)+\int(du)/(2(u+2))-\int(du)/(u+1)`

`=(log(u))/(2)+(log(u+2))/(2)-log(u+1)}`
4. back-substitute u=e^x

`=(log(e^x))/(2)+(log(e^x+2))/(2)-log(e^x+1)}`

`=(x)/(2)+(log(e^x+2))/(2)-log(e^x+1)}`

Note: log(x) stands for natural log, and NOT log10(x)