$\int\frac{1}{(x+5)^{7/8}(x-3)^{9/8}}dx$ is equal to:

Question

Accepted Answer

To solve the integral $\int\frac{1}{(x+5)^{7/8}(x-3)^{9/8}}dx$, we need to find a function whose derivative matches the integrand. The correct answer is given as Option 3: $-\left(\frac{x+5}{x-3}\right)^{1/8} + c$.

### Explanation for Option 3:

1. **Substitution Method**: 
   - Consider the substitution $ u = \left(\frac{x+5}{x-3}\right)^{1/8} $. This implies that $ u^8 = \frac{x+5}{x-3} $.
   - Differentiating both sides with respect to $ x $, we get:
     $$
     8u^7 \cdot \frac{d}{dx}\left(\frac{x+5}{x-3}\right) = \frac{d}{dx}(u^8)
     $$
   - The derivative of $\frac{x+5}{x-3}$ is $\frac{(x-3) - (x+5)}{(x-3)^2} = \frac{-8}{(x-3)^2}$.
   - Therefore, $ \frac{d}{dx}\left(\frac{x+5}{x-3}\right) = \frac{-8}{(x-3)^2} $.

2. **Integration**:
   - The integral becomes:
     $$
     \int \frac{1}{(x+5)^{7/8}(x-3)^{9/8}} dx = \int -\frac{8}{(x-3)^2} \cdot \frac{1}{u^7} \cdot \frac{1}{8u^7} du
     $$
   - Simplifying, we get:
     $$
     \int -\frac{1}{u} du = -\ln|u| + c
     $$
   - Substituting back for $ u $, we have:
     $$
     -\ln\left|\left(\frac{x+5}{x-3}\right)^{1/8}\right| + c = -\left(\frac{x+5}{x-3}\right)^{1/8} + c
     $$

### Explanation for Why Other Options Are Incorrect:

1. **Option 1: $\left(\frac{x+5}{x-3}\right)^{1/8} + c$**:
   - This option lacks the negative sign, which is crucial because the derivative of the function inside the integral involves a negative factor due to the subtraction in the denominator. The correct integration process introduces a negative sign, which is missing here.

2. **Option 2: $\left(\frac{x+4}{x-4}\right)^{1/8} + c$**:
   - This option uses incorrect expressions in the numerator and denominator. The original integral involves $(x+5)$ and $(x-3)$, not $(x+4)$ and $(x-4)$. Therefore, this substitution does not match the structure of the integrand.

3. **Option 4: $-\left(\frac{x+4}{x-4}\right)^{1/8} + c$**:
   - Similar to Option 2, this option uses incorrect expressions in the numerator and denominator. The substitution does not align with the original integral, making it incorrect.

In conclusion, Option 3 is the correct answer because it accurately reflects the integration process and the necessary substitution, including the correct negative sign derived from the differentiation of the function inside the integral.