What is the integral of e^x(ln(x+1)+1/(x+1)) ?

The integral of e^x(ln(x+1)+1/(x+1)) is e^xln(x+1)+C

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integral of e^x(ln(x+1)+1/(x+1))

\int e^{x} (ln (x + 1) + \frac{1}{x + 1}) d x

e^{x} ln (x + 1) + C

Solution steps

\int e^{x} (ln (x + 1) + \frac{1}{x + 1}) d x

Expand

e^{x} (ln (x + 1) + \frac{1}{x + 1}) : e^{x} ln (x + 1) + \frac{e ^{x}}{x + 1}

= \int e^{x} ln (x + 1) + \frac{e ^{x}}{x + 1} 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^{x} ln (x + 1) d x + \int \frac{e ^{x}}{x + 1} d x

\int e^{x} ln (x + 1) d x = e^{x} ln (x + 1) - \frac{1}{e} Ei (x + 1)

\int \frac{e ^{x}}{x + 1} d x = \frac{1}{e} Ei (x + 1)

= e^{x} ln (x + 1) - \frac{1}{e} Ei (x + 1) + \frac{1}{e} Ei (x + 1)

Add similar elements:

- \frac{1}{e} Ei (x + 1) + \frac{1}{e} Ei (x + 1) = 0

= e^{x} ln (x + 1)

Add a constant to the solution

= e^{x} ln (x + 1) + C

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What is the integral of e^x(ln(x+1)+1/(x+1)) ?
The integral of e^x(ln(x+1)+1/(x+1)) is e^xln(x+1)+C