The integral shown in the image is:

$\int \left( \frac{6}{x^2} - \frac{7}{x^4} \right) dx$

Solution:

We can split the integral into two separate terms:

$\int \left( \frac{6}{x^2} - \frac{7}{x^4} \right) dx = \int \frac{6}{x^2} dx - \int \frac{7}{x^4} dx$

Step 1: Rewrite the terms with negative exponents

$\frac{6}{x^2} = 6x^{-2}, \quad \frac{7}{x^4} = 7x^{-4}$

Thus: $\int \left( \frac{6}{x^2} - \frac{7}{x^4} \right) dx = \int 6x^{-2} dx - \int 7x^{-4} dx$

Step 2: Apply the power rule of integration

For any $x^n$, the integral is: \int x^n dx = \frac{x^{n+1}}{n+1} \quad \text{(for n \neq -1)}

First term: $\int 6x^{-2} dx = 6 \cdot \frac{x^{-2+1}}{-2+1} = 6 \cdot \frac{x^{-1}}{-1} = -\frac{6}{x}$

Second term: $\int 7x^{-4} dx = 7 \cdot \frac{x^{-4+1}}{-4+1} = 7 \cdot \frac{x^{-3}}{-3} = -\frac{7}{3x^3}$

Step 3: Combine the results

$\int \left( \frac{6}{x^2} - \frac{7}{x^4} \right) dx = -\frac{6}{x} - \frac{7}{3x^3} + C$

Where $C$ is the constant of integration.

Final Answer:

$-\frac{6}{x} - \frac{7}{3x^3} + C$

Let me know if you'd like further clarification or have any questions! Here are some related problems to consider:

1. What is the integral of $\frac{a}{x^n}$ where $n > 0$ and $a$ is constant?
2. How does the power rule for integration differ from the power rule for differentiation?
3. How would this problem change if there were definite bounds for the integral?
4. What happens if $n = -1$ in the integral $\int x^n dx$?
5. How can you apply similar techniques to integrate trigonometric or logarithmic functions?

Tip: Always rewrite the terms with fractional or negative exponents before integrating—it simplifies applying the power rule!