Question

Evaluate ∫1+4xexxdx. Here C is the constant of integration. Use abs(x) to denote |x|.

Answer 1

Given the integral:

`\int(1+4xe^x)/(x)`

You can evaluate it as follows:

1. Separate it into two integrals with the same denominator:

`=\int(1)/(x)dx+\int(4xe^x)/(x)dx`

2. Write the constants outside the integral:

`=\int(1)/(x)dx+4\int(xe^x)/(x)dx`

3. Since:

`(x)/(x)=1`

You can keep simplifying:

`=\int(1)/(x)dx+4\int e^xdx`

4. Integrate by applying these Integration Rules:

`\begin{gathered} \int e^xdx=e^x \\ \\ \int(1)/(x)dx=ln|x|+C \end{gathered}`

You get:

`=ln|x|+4e^x+C`

Hence, the answer is:

`=ln|x|+4e^x+C`

Or:

`=ln(abs(x))+4e^x+C`