Given the integral:
[tex]\int\frac{1+4xe^x}{x}[/tex]
You can evaluate it as follows:
1. Separate it into two integrals with the same denominator:
[tex]=\int\frac{1}{x}dx+\int\frac{4xe^x}{x}dx[/tex]
2. Write the constants outside the integral:
[tex]=\int\frac{1}{x}dx+4\int\frac{xe^x}{x}dx[/tex]
3. Since:
[tex]\frac{x}{x}=1[/tex]
You can keep simplifying:
[tex]=\int\frac{1}{x}dx+4\int e^xdx[/tex]
4. Integrate by applying these Integration Rules:
[tex]\begin{gathered} \int e^xdx=e^x \\ \\ \int\frac{1}{x}dx=ln|x|+C \end{gathered}[/tex]
You get:
[tex]=ln|x|+4e^x+C[/tex]
Hence, the answer is:
[tex]=ln|x|+4e^x+C[/tex]
Or:
[tex]=ln(abs(x))+4e^x+C[/tex]
