Key Concepts and Formulas

• Substitution Method: A technique to simplify integrals by substituting a function of $x$ with a new variable, $t$, and adjusting the differential accordingly.
• Power Rule for Integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$.

Step-by-Step Solution

Step 1: Rewrite the Integral

We want to simplify the integral by strategically manipulating the terms. Notice that the exponents of $(x+4)$ and $(x-3)$ add up to $\frac{8}{7} + \frac{6}{7} = \frac{14}{7} = 2$. We can rewrite the integral as follows:

$\int \frac{dx}{(x+4)^{\frac{8}{7}}(x-3)^{\frac{6}{7}}} = \int \frac{dx}{(x+4)^2 (\frac{x-3}{x+4})^{\frac{6}{7}}}$

The reason for doing this is to create a term of the form $\frac{x-3}{x+4}$ inside the integral, which will allow us to use the substitution method effectively.

Step 2: Apply the Substitution

Let $t = \frac{x-3}{x+4}$. Then, we need to find $dt$ in terms of $dx$. Differentiating $t$ with respect to $x$:

$\frac{dt}{dx} = \frac{(x+4)(1) - (x-3)(1)}{(x+4)^2} = \frac{x+4-x+3}{(x+4)^2} = \frac{7}{(x+4)^2}$

Therefore,

$dt = \frac{7}{(x+4)^2} dx$

which implies

$\frac{dx}{(x+4)^2} = \frac{1}{7} dt$

Step 3: Substitute into the Integral

Now we can substitute $t$ and $dt$ into the integral:

$\int \frac{dx}{(x+4)^2 (\frac{x-3}{x+4})^{\frac{6}{7}}} = \int \frac{\frac{1}{7} dt}{t^{\frac{6}{7}}} = \frac{1}{7} \int t^{-\frac{6}{7}} dt$

Step 4: Integrate with Respect to t

Using the power rule for integration:

$\frac{1}{7} \int t^{-\frac{6}{7}} dt = \frac{1}{7} \cdot \frac{t^{-\frac{6}{7} + 1}}{-\frac{6}{7} + 1} + C = \frac{1}{7} \cdot \frac{t^{\frac{1}{7}}}{\frac{1}{7}} + C = t^{\frac{1}{7}} + C$

Step 5: Substitute Back for x

Finally, substitute $t = \frac{x-3}{x+4}$ back into the expression:

$t^{\frac{1}{7}} + C = \left(\frac{x-3}{x+4}\right)^{\frac{1}{7}} + C$

Common Mistakes & Tips

• Algebraic Manipulation: Be careful when manipulating the exponents and terms inside the integral. Double-check your algebra to avoid errors.
• Substitution: Choosing the right substitution is crucial. In this case, recognizing the form $\frac{x-3}{x+4}$ simplified the integral significantly.
• Constant of Integration: Don't forget to add the constant of integration, $C$, after evaluating the indefinite integral.

Summary

We simplified the given integral by rewriting it to expose the term $\frac{x-3}{x+4}$. We then used the substitution method, letting $t = \frac{x-3}{x+4}$, which simplified the integral into a form where we could apply the power rule for integration. After integrating with respect to $t$, we substituted back to express the result in terms of $x$. The final result is $\left(\frac{x-3}{x+4}\right)^{\frac{1}{7}} + C$.

Final Answer

The final answer is $\boxed{\left(\frac{x-3}{x+4}\right)^{\frac{1}{7}} + C}$, which corresponds to option (B).