Example 1: Indefinite Integral ln(x)

$\int \frac{9}{x}\mathit{ dx}$

Step 1: Simplify and look for algebraic rewrites.

None in this example.

$\int \frac{9}{x}\mathit{ dx}$

Step 2: Identify any term(s) that includes the base case $\frac{n}{x^1}$.

Here you have 1 chunk, and it is the base case $\frac{9}{x}$.

$\int \frac{9}{x}\mathit{ dx}$

Step 3: Take the antiderivativeof the $\frac{n}{x^1}$ special cases using their specific Recipe.

Chunk 1: $\frac{9}{x}$

n = 9

$\int \frac{9}{x}\mathit{ dx}=\mathit{9}\mathit{\bullet }\mathit{ln}\left|x\right|\mathit{+}\mathit{C}$

DO NOT FORGET THE +C

Final Result Meaning: Remember the Definite Integral will always provide you a definite value , and the Indefinite Integral provides you a family of solutions .

The antiderivative of the equation $f‘\left(x\right)=\frac{9}{x}$is the family of graphs $f\left(x\right)=\mathit{9}\mathit{\bullet }\mathit{ln}\left|x\right|\mathit{+}\mathit{C}$.

All graphs of the form $f\left(x\right)=\mathit{9}\mathit{\bullet }\mathit{ln}\left|x\right|\mathit{+}\mathit{C}$ have the same derivative (instantaneous rate of change), $f‘\left(x\right)=\frac{9}{x}$, at any x-value .

The only difference between any of original graphs, $f\left(x\right)=\mathit{9}\mathit{\bullet }\mathit{ln}\left|x\right|\mathit{+}\mathit{C}$, is just a vertical shift of +C .

In the graph above are examples of 3-different C-values. You can see the instantaneous rate of change ( slope ) is the same at every x-value between all the different examples, no matter the +C .