To find the integral of the function [tex]\( f(x) = -\frac{5}{6} x^{\frac{1}{3}} - 2 x^{\frac{4}{5}} - 9 \)[/tex] with respect to [tex]\( x \)[/tex], we will integrate each term of the function separately.
Let's break down the integral step-by-step:
[tex]\[
\int \left(-\frac{5}{6} x^{\frac{1}{3}} - 2 x^{\frac{4}{5}} - 9\right) dx
\][/tex]
### Step 1: Integrate [tex]\(-\frac{5}{6} x^{\frac{1}{3}}\)[/tex]
Using the power rule for integration, [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex], we integrate [tex]\(-\frac{5}{6} x^{\frac{1}{3}}\)[/tex]:
[tex]\[
-\frac{5}{6} \int x^{\frac{1}{3}} \, dx = -\frac{5}{6} \left(\frac{x^{\frac{1}{3} + 1}}{\frac{1}{3} + 1}\right)
\][/tex]
Simplify the exponent:
[tex]\[
\frac{1}{3} + 1 = \frac{1}{3} + \frac{3}{3} = \frac{4}{3}
\][/tex]
Thus, the integral becomes:
[tex]\[
-\frac{5}{6} \left(\frac{x^{\frac{4}{3}}}{\frac{4}{3}}\right) = -\frac{5}{6} \cdot \frac{3}{4} x^{\frac{4}{3}} = -\frac{5}{8} x^{\frac{4}{3}}
\][/tex]
### Step 2: Integrate [tex]\(-2 x^{\frac{4}{5}}\)[/tex]
Again, using the power rule for integration:
[tex]\[
-2 \int x^{\frac{4}{5}} \, dx = -2 \left(\frac{x^{\frac{4}{5} + 1}}{\frac{4}{5} + 1}\right)
\][/tex]
Simplify the exponent:
[tex]\[
\frac{4}{5} + 1 = \frac{4}{5} + \frac{5}{5} = \frac{9}{5}
\][/tex]
Thus, the integral becomes:
[tex]\[
-2 \left(\frac{x^{\frac{9}{5}}}{\frac{9}{5}}\right) = -2 \cdot \frac{5}{9} x^{\frac{9}{5}} = -\frac{10}{9} x^{\frac{9}{5}}
\][/tex]
### Step 3: Integrate [tex]\(-9\)[/tex]
This is a straightforward application of the power rule for integration:
[tex]\[
\int -9 \, dx = -9x
\][/tex]
### Combine the Results
Now, we combine the results of each integral:
[tex]\[
-\frac{5}{8} x^{\frac{4}{3}} - \frac{10}{9} x^{\frac{9}{5}} - 9x + C
\][/tex]
Therefore, the integral of the given function is:
[tex]\[
\int \left(-\frac{5}{6} x^{\frac{1}{3}} - 2 x^{\frac{4}{5}} - 9\right) dx = -\frac{10}{9} x^{\frac{9}{5}} - \frac{5}{8} x^{\frac{4}{3}} - 9x + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
