`int(dx)/((x+3)^(8//7)(x-2)^(6//7))` is equal to

Options

A. `(5)/(7)((x-2)/(x+3))^(1//7)+c`
B. `(7)/(5)((x+3)/(x-2))^(1//7)+c`
C. `(7)/(5)((x-2)/(x+3))^(-1//7)+c`
D. none of these

Answer

To solve the integral \( I = \int \frac{dx}{(x+3)^{\frac{8}{7}}(x-2)^{\frac{6}{7}}} \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{dx}{(x+3)^{\frac{8}{7}}(x-2)^{\frac{6}{7}}} \]
To simplify the integration, we can multiply and divide the integrand by \( (x-2)^{\frac{8}{7}} \):
\[ I = \int \frac{(x-2)^{\frac{8}{7}} \, dx}{(x+3)^{\frac{8}{7}}(x-2)^{\frac{8}{7}}(x-2)^{\frac{6}{7}}} \]
This gives us:
\[ I = \int \frac{(x-2)^{\frac{8}{7}} \, dx}{(x+3)^{\frac{8}{7}}(x-2)^{\frac{14}{7}}} \]
Now we can combine the powers of \( (x-2) \):
\[ I = \int \frac{dx}{(x+3)^{\frac{8}{7}}(x-2)^{2}} \]

### Step 2: Substitution
Let:
\[ t = \frac{x+3}{x-2} \]
Then, we differentiate both sides with respect to \( x \):
\[ dt = \frac{(x-2)(1) - (x+3)(1)}{(x-2)^2} \, dx = \frac{-5}{(x-2)^2} \, dx \]
This implies:
\[ dx = -\frac{5}{(x-2)^2} dt \]

### Step 3: Substitute in the Integral
Substituting \( dx \) and \( t \) into the integral:
\[ I = \int \frac{-\frac{5}{(x-2)^2} dt}{\left(\frac{x+3}{x-2}\right)^{\frac{8}{7}}} \]
This simplifies to:
\[ I = -5 \int t^{-\frac{8}{7}} dt \]

### Step 4: Integrate
Now we can integrate:
\[ I = -5 \cdot \left( \frac{t^{-\frac{8}{7}+1}}{-\frac{8}{7}+1} \right) + C \]
Calculating \( -\frac{8}{7}+1 = \frac{-8+7}{7} = -\frac{1}{7} \):
\[ I = -5 \cdot \left( \frac{t^{-\frac{1}{7}}}{\frac{-1}{7}} \right) + C = 35 t^{-\frac{1}{7}} + C \]

### Step 5: Substitute Back for \( t \)
Now substitute back \( t = \frac{x+3}{x-2} \):
\[ I = 35 \left(\frac{x-2}{x+3}\right)^{\frac{1}{7}} + C \]

### Final Answer
Thus, the final answer for the integral is:
\[ I = \frac{35 (x-2)^{\frac{1}{7}}}{(x+3)^{\frac{1}{7}}} + C \]