(d) $$ \int \pi x^{\frac{5}{6}} \mathrm{~d} x $$

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UpStudy AI · Accepted Answer

1. Factor out the constant:
   
   $$
   \int \pi x^{\frac{5}{6}} \, dx = \pi \int x^{\frac{5}{6}} \, dx
   $$

2. Use the power rule for integration:
   
   $$
   \int x^a \, dx = \frac{x^{a+1}}{a+1} + C \quad \text{(for } a \neq -1\text{)}
   $$
   
   Here, $$ a = \frac{5}{6} $$.

3. Calculate $$ a + 1 $$:
   
   $$
   \frac{5}{6} + 1 = \frac{5}{6} + \frac{6}{6} = \frac{11}{6}
   $$

4. Substitute into the formula:
   
   $$
   \int x^{\frac{5}{6}} \, dx = \frac{x^{\frac{11}{6}}}{\frac{11}{6}} + C
   $$

5. Multiply by the constant $$ \pi $$:
   
   $$
   \pi \cdot \frac{x^{\frac{11}{6}}}{\frac{11}{6}} = \pi \cdot \frac{6}{11} x^{\frac{11}{6}} = \frac{6\pi}{11} x^{\frac{11}{6}} 
   $$

6. Write the final result with the constant of integration:
   
   $$
   \frac{6\pi}{11} x^{\frac{11}{6}} + C
   $$
The integral of $$ \pi x^{\frac{5}{6}} $$ with respect to $$ x $$ is $$ \frac{6\pi}{11} x^{\frac{11}{6}} + C $$.

(d)

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