Example 1: Definite Integral ln(x)

$\underset{3}{\overset{7}{\int }}\frac{9}{x}\mathit{ dx}$

Step 1: Simplify and look for algebraic rewrites.

None in this example.

$\underset{3}{\overset{7}{\int }}\frac{9}{x}\mathit{ dx}$

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 $\frac{9}{x}$.

$\underset{3}{\overset{7}{\int }}\frac{9}{x}\mathit{ dx}$

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

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

n = 9

$\underset{3}{\overset{7}{\int }}\frac{9}{x}\mathit{ dx}\mathit{=}{\mathit{9}\mathit{\bullet }\mathit{ln}\left|x\right||}_{x=3}^{x=7}$

Step 4 ( Definite Integral ONLY ): Evaluate the antiderivative result using the Top – Bottom method.

$\underset{3}{\overset{7}{\int }}\frac{7}{x}\mathit{ dx}\mathit{=}{\mathit{7}\mathit{\bullet }\mathit{ln}\left|x\right||}_{x=3}^{x=7}$

$=\left(\mathit{7}\mathit{\bullet }\mathit{ln}\left|\left(7\right)\right|\right)–\left(\mathit{9}\mathit{\bullet }\mathit{ln}\left|\left(3\right)\right|\right)\approx 7.625$

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 Net Area between the curve $f\left(x\right)=\frac{9}{x}$ and the x-axis on the x-interval [ 3 , 7 ] is 7.625 .

Since the final result is positive, you know without even seeing the graph that there is more area above the x-axis than below it.