The integral shown in the image is:

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

Solution:

1. Split the integral into two parts: $\int \frac{6}{x^2} dx - \int \frac{7}{x^4} dx$

2. Rewrite the fractions as powers of $x$: $\frac{6}{x^2} = 6x^{-2}, \quad \frac{7}{x^4} = 7x^{-4}$

   So the integral becomes: $\int 6x^{-2} dx - \int 7x^{-4} dx$

3. Use the power rule for integration: \int x^n dx = \frac{x^{n+1}}{n+1} \quad \text{(for n \neq -1)}.

   • For $6x^{-2}$: $n = -2$ $\int 6x^{-2} dx = 6 \cdot \frac{x^{-1}}{-1} = -\frac{6}{x}$

   • For $7x^{-4}$: $n = -4$ $\int 7x^{-4} dx = 7 \cdot \frac{x^{-3}}{-3} = -\frac{7}{3x^3}$

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

Final Answer:

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

Let me know if you would like further clarifications or details on any step!

Related Questions:

1. What is the power rule for integration and how does it work?
2. How would you evaluate this integral with definite limits?
3. What is the significance of the constant $C$ in indefinite integrals?
4. How do we handle integrals where the denominator has higher-degree polynomials?
5. Could this problem be solved using substitution? Why or why not?

Tip:

When integrating terms like $\frac{a}{x^n}$, always rewrite them as $a \cdot x^{-n}$ to apply the power rule easily.