Solve ∫ e^2t-1/e^t-1dt

$$\int \frac { e ^ { 2 t } - 1 } { e ^ { t } - 1 } d t$$

Evaluate

$$t+e^{t}+С$$

Solution Steps

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

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

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

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

Integrate the sum term by term.

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

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

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

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

$$e^{t}+t$$

If F\left(t\right) is an antiderivative of f\left(t\right), then the set of all antiderivatives of f\left(t\right) is given by F\left(t\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$e^{t}+t+С$$

Differentiate w.r.t. t

$$e^{t}+1$$