Question

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

Answer 1

Given the integral:

$\int\frac{1+4xe^x}{x}$

You can evaluate it as follows:

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

$=\int\frac{1}{x}dx+\int\frac{4xe^x}{x}dx$

2. Write the constants outside the integral:

$=\int\frac{1}{x}dx+4\int\frac{xe^x}{x}dx$

3. Since:

$\frac{x}{x}=1$

You can keep simplifying:

$=\int\frac{1}{x}dx+4\int e^xdx$

4. Integrate by applying these Integration Rules:

$\begin{gathered} \int e^xdx=e^x \\ \\ \int\frac{1}{x}dx=ln|x|+C \end{gathered}$

You get:

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

Hence, the answer is:

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

Or:

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