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int(1)/(e^(-x)-1)dx

$\int \frac{1}{e^{-x}-1} d x$

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Evaluate definite integrals using the chain rule

Full solution

Q.

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

Simplify integrand: Step $1$ : Simplify the integrand. $\newline$ Rewrite the integrand using the property of exponents: $e^{-x} = 1/e^x$ . $\newline$ So, $1/(e^{-x} - 1)$ becomes $1/(1/e^x - 1)$ . $\newline$ Simplify further: $1/(1/e^x - 1) = e^x/(1 - e^x)$ .
Substitute and simplify: Step $2$ : Substitute to simplify the integral. $\newline$ Let $u = 1 - e^x$ , then $du = -e^x dx$ . $\newline$ Rearrange for $dx$ : $dx = -\frac{du}{e^x}$ . $\newline$ Substitute into the integral: $\int \frac{e^x}{1 - e^x} dx = \int -\frac{du}{u}$ .
Integrate with new variable: Step $3$ : Integrate using the new variable. $\newline$ The integral $\int -\frac{du}{u}$ is $-\ln|u| + C$ .
Substitute back to x: Step $4$ : Substitute back to $x$ . $\newline$ Since $u = 1 - e^x$ , the integral becomes $-\ln|1 - e^x| + C$ .

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