What is the integral of (xe^{-x^2}+(e^x)/(e^x+3)) ?

The integral of (xe^{-x^2}+(e^x)/(e^x+3)) is -1/2 e^{-x^2}+ln|e^x+3|+C

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integral of (xe^{-x^2}+(e^x)/(e^x+3))

\int (x e^{- x^{2}} + \frac{e ^{x}}{e ^{x} + 3}) d x

- \frac{1}{2} e^{- x^{2}} + ln ∣ e^{x} + 3 ∣ + C

Solution steps

\int x e^{- x^{2}} + \frac{e ^{x}}{e ^{x} + 3} 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 x e^{- x^{2}} d x + \int \frac{e ^{x}}{e ^{x} + 3} d x

\int x e^{- x^{2}} d x = - \frac{1}{2} e^{- x^{2}}

\int \frac{e ^{x}}{e ^{x} + 3} d x = ln ∣ e^{x} + 3 ∣

= - \frac{1}{2} e^{- x^{2}} + ln ∣ e^{x} + 3 ∣

Add a constant to the solution

= - \frac{1}{2} e^{- x^{2}} + ln ∣ e^{x} + 3 ∣ + C

What is the integral of (xe^{-x^2}+(e^x)/(e^x+3)) ?
The integral of (xe^{-x^2}+(e^x)/(e^x+3)) is -1/2 e^{-x^2}+ln|e^x+3|+C