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Evaluate
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Differentiate w.r.t. x
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\int \frac{\left(e^{x}-1\right)\left(e^{x}+1\right)}{e^{x}-1}\mathrm{d}x
Factor the expressions that are not already factored in \frac{e^{2x}-1}{e^{x}-1}.
\int e^{x}+1\mathrm{d}x
Cancel out e^{x}-1 in both numerator and denominator.
\int e^{x}\mathrm{d}x+\int 1\mathrm{d}x
Integrate the sum term by term.
e^{x}+\int 1\mathrm{d}x
Use \int e^{x}\mathrm{d}x=e^{x} from the table of common integrals to obtain the result.
e^{x}+x
Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.
e^{x}+x+С
If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.