The integral in the image is:

$\int 3x^{-4} \left( 2 + 4x^{-3} \right)^{-7} \, dx$

Step-by-Step Solution

We can solve this integral using substitution. Let’s define:

$u = 2 + 4x^{-3}$

Now, differentiate $u$ with respect to $x$:

$\frac{du}{dx} = -12x^{-4}$

This implies:

$du = -12x^{-4} \, dx$

Now, rewrite the integral:

$\int 3x^{-4} \left( 2 + 4x^{-3} \right)^{-7} \, dx = \int 3x^{-4} u^{-7} \, dx$

Since $du = -12x^{-4} \, dx$, we can replace $3x^{-4} \, dx$ as follows:

$3x^{-4} \, dx = -\frac{1}{4} \, du$

Thus, the integral becomes:

$-\frac{1}{4} \int u^{-7} \, du$

Now, integrate $u^{-7}$:

$-\frac{1}{4} \left( \frac{u^{-6}}{-6} \right) = \frac{1}{24} u^{-6}$

Substitute back $u = 2 + 4x^{-3}$:

$\frac{1}{24} \left( 2 + 4x^{-3} \right)^{-6} + C$

Final Answer:

$\int 3x^{-4} \left( 2 + 4x^{-3} \right)^{-7} \, dx = \frac{1}{24} \left( 2 + 4x^{-3} \right)^{-6} + C$

Would you like a more detailed breakdown or have any questions?

Here are 5 related questions for further exploration:

1. What is the substitution method in calculus, and how does it work?
2. How do you determine the correct substitution for more complex integrals?
3. Can substitution be used for definite integrals, and how do the limits change?
4. What are other techniques for solving difficult integrals, like by parts?
5. How can we interpret the result of this integral geometrically?

Tip: Always check if simplifying an expression through substitution or algebraic manipulation can make the integral easier to solve.