What is the integral of e^{-knx}+e^{-ln(x)} ?

The integral of e^{-knx}+e^{-ln(x)} is -1/(kn)e^{-knx}+ln|x|+C

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integral of e^{-knx}+e^{-ln(x)}

\int e^{- kn x} + e^{- l n (x)} d x

- \frac{1}{kn} e^{- kn x} + ln ∣ x ∣ + C

Solution steps

\int e^{- kn x} + e^{- l n (x)} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{- kn x} d x + \int e^{- l n (x)} d x

\int e^{- kn x} d x = - \frac{1}{kn} e^{- kn x}

\int e^{- l n (x)} d x = ln ∣ x ∣

= - \frac{1}{kn} e^{- kn x} + ln ∣ x ∣

Add a constant to the solution

= - \frac{1}{kn} e^{- kn x} + ln ∣ x ∣ + C

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What is the integral of e^{-knx}+e^{-ln(x)} ?
The integral of e^{-knx}+e^{-ln(x)} is -1/(kn)e^{-knx}+ln|x|+C