Solve ∫ (2{texttt{e}}^2+3/x) wrt x

$$\int{ (2 { \texttt{e} }^{ 2 } + \frac{ 3 }{ x } ) }d x$$

Evaluate

$$3\ln(|x|)+2e^{2}x+С$$

Solution Steps

Integrate the sum term by term.

$$\int 2e^{2}\mathrm{d}x+\int \frac{3}{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int e^{2}\mathrm{d}x+3\int \frac{1}{x}\mathrm{d}x$$

Find the integral of e^{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$2e^{2}x+3\int \frac{1}{x}\mathrm{d}x$$

Use \int \frac{1}{x}\mathrm{d}x=\ln(|x|) from the table of common integrals to obtain the result.

$$2e^{2}x+3\ln(|x|)$$

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.

$$2e^{2}x+3\ln(|x|)+С$$

Differentiate w.r.t. x

$$2e^{2}+\frac{3}{x}$$