Chapter 5.1, Problem 1ED

Explore and Discuss 1

Formulas 1, 2, and 3 do not provide a formula for the indefinite integral of the function In x. Show that if x > 0, then

     ${\displaystyle \int \text{ln}}xdx=x\text{ln}x-x+C$

by differentiating the right-hand side.

FORMULAS Indefinite Integrals of Basic Functions

For C a constant,

1.

     $\begin{array}{cc}{\displaystyle \int x^{n}dx=\frac{x^{n+1}}{n+1}}+C& n\ne -1\end{array}$

2.

     ${\displaystyle \int e^{x}dx=e^x+C}$

3.

     $\begin{array}{cc}{\displaystyle \int \frac{1}{x}dx=\text{ln}\left|x\right|}+C,& x\ne 0\end{array}$