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Evaluate: int(e^x)/(e^(2x)+4e^x+3)dx...

Evaluate: `int(e^x)/(e^(2x)+4e^x+3)dx`

Text Solution

Verified by Experts

The correct Answer is:
`(1)/(4)"log"|(e^(x)+1)/(e^(x)+5)|+C`
`I=int(e^(x))/(e^(2x)+6e^(x)+5)dx=int(e^(x))/((e^(x))^(2)+6e^(x)+5)dx`
` "Let "e^(x)=t " or " e^(x)dx=dt`
` :. I=int(dt)/(t^(2)+6t+5)`
`=int(1)/((t+3)^(2)-2^(2))dt`
`=(1)/(2xx2)"log"|(t+3-2)/(t+3+2)|+C=(1)/(4)"log"|(e^(x)+1)/(e^(x)+5)|+C`
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