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Question
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
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Solution
Let I = `int "e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`= int e^x/sqrt((e^x)^2 + 4e^x + 13)` dx
Put ex = t
∴ ex dx = dt
∴ I = `(dt)/(sqrt(t^2 + 4t + 13))`
`= int 1/sqrt(t^2 + 4t + 4 - 4 + 13)` dt
`= int 1/(sqrt((t + 2)^2 + 9))` dt
`= int 1/(sqrt((t + 2)^2 + (3)^2))` dt
`= log |t + 2 + sqrt((t + 2)^2 + (3)^2)|` + c
`= log |(t + 2) + sqrt(t^2 + 4t + 13)| + c`
∴ I = `log |(e^x + 2) + sqrt(e^(2x) + 4e^x + 13)| + c`
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