https://file.aiquickdraw.com/mathbot/file/17263246280306ja4bnk1.jpg

The integral in the image is:

\[
\int \left( 3x^6 - \frac{2}{x^3} \right) dx
\]

We can solve this by integrating each term separately.

### Step 1: Integrating \( 3x^6 \)

The general rule for integrating powers of \(x\) is:

\[
\int x^n dx = \frac{x^{n+1}}{n+1} + C
\]

So, for \( 3x^6 \), we apply the formula:

\[
\int 3x^6 dx = 3 \cdot \frac{x^{6+1}}{6+1} = \frac{3x^7}{7}
\]

### Step 2: Integrating \( -\frac{2}{x^3} \)

We can rewrite \( \frac{2}{x^3} \) as \( 2x^{-3} \) and then apply the power rule:

\[
\int 2x^{-3} dx = 2 \cdot \frac{x^{-3+1}}{-3+1} = 2 \cdot \frac{x^{-2}}{-2} = -\frac{2}{2}x^{-2} = -x^{-2}
\]

### Step 3: Combining both results

Now, we combine the two integrals:

\[
\int \left( 3x^6 - \frac{2}{x^3} \right) dx = \frac{3x^7}{7} - x^{-2} + C
\]

Thus, the final answer is:

\[
\boxed{\frac{3x^7}{7} - \frac{1}{x^2} + C}
\]

Would you like further details on this solution or clarification?

### Related questions:
1. How do you integrate functions with fractional exponents?
2. How does the power rule apply to negative exponents in integrals?
3. What is the integral of \(x^{-1}\)?
4. How do you handle integration when faced with more complex functions?
5. What are the applications of integrals involving polynomials in physics?

### Tip:
When integrating, always check if you can simplify expressions before applying rules—this helps avoid mistakes with exponents and constants.

Learn how to solve the integral of (3x^6 - 2/x^3) with detailed steps, applying the power rule of integration. This problem covers fundamental concepts in calculus, suitable for high school students in Grades 11-12.

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