Solve ∫ e^2x-1/e^x-1

$$\int \frac { e ^ { 2 x } - 1 } { e ^ { x } - 1 }$$

Evaluate

$$x+e^{x}+С$$

Solution Steps

Factor the expressions that are not already factored in \frac{e^{2x}-1}{e^{x}-1}.

$$\int \frac{\left(e^{x}-1\right)\left(e^{x}+1\right)}{e^{x}-1}\mathrm{d}x$$

Cancel out e^{x}-1 in both numerator and denominator.

$$\int e^{x}+1\mathrm{d}x$$

Integrate the sum term by term.

$$\int e^{x}\mathrm{d}x+\int 1\mathrm{d}x$$

Use \int e^{x}\mathrm{d}x=e^{x} from the table of common integrals to obtain the result.

$$e^{x}+\int 1\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$e^{x}+x$$

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.

$$e^{x}+x+С$$

Differentiate w.r.t. x

$$e^{x}+1$$