Solve ∫ 1/1+ARCVe^Av+1dr

$$\int \frac { 1 } { 1 + A R C V } e ^ { A v + 1 } d r$$

Evaluate

$$\frac{re^{Av+1}}{ACRV+1}+С$$

$$A\neq -\frac{1}{CRV}\text{ or }V=0\text{ or }C=0\text{ or }R=0$$

Solution Steps

Find the integral of \frac{e^{Av+1}}{1+ARCV} using the table of common integrals rule \int a\mathrm{d}r=ar.

$$\frac{e^{Av+1}}{1+ARCV}r$$

Simplify.

$$\frac{e^{Av+1}r}{1+ARCV}$$

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

$$\begin{matrix}\frac{e^{Av+1}r}{1+ARCV}+С_{3},&\end{matrix}$$

Differentiate w.r.t. A

$$\frac{r\left(ACRVv+v-CRV\right)e^{Av+1}}{\left(ACRV+1\right)^{2}}$$