`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 can rewrite the integral in a more manageable form by multiplying and dividing by \((x-2)^{\frac{1}{7}}\):

\[ I = \int \frac{(x-2)^{\frac{1}{7}}}{(x+3)^{\frac{8}{7}}(x-2)^{\frac{6}{7}}(x-2)^{\frac{1}{7}}} \, dx = \int \frac{(x-2)^{\frac{1}{7}}}{(x+3)^{\frac{8}{7}}(x-2)^{\frac{6}{7}}} \, dx \]

### Step 2: Simplify the Integral
This simplifies to:

\[ I = \int \frac{1}{(x+3)^{\frac{8}{7}}} \cdot (x-2)^{\frac{1}{7}} \, dx \]

### Step 3: Use Substitution
Let \( t = x - 2 \), then \( dx = dt \) and \( x = t + 2 \). Substitute into the integral:

\[ I = \int \frac{(t)^{\frac{1}{7}}}{(t + 2 + 3)^{\frac{8}{7}}} \, dt = \int \frac{t^{\frac{1}{7}}}{(t + 5)^{\frac{8}{7}}} \, dt \]

### Step 4: Further Simplification
Now we can express the integral as:

\[ I = \int t^{\frac{1}{7}} (t + 5)^{-\frac{8}{7}} \, dt \]

### Step 5: Use Integration by Parts or Beta Function
This integral can be evaluated using integration techniques such as integration by parts or recognizing it as a Beta function. However, we will proceed with the integration directly.

### Step 6: Integrate
Using the formula for integration, we can integrate:

\[ I = \frac{7}{5} \cdot \frac{t^{\frac{8}{7}}}{(t + 5)^{\frac{8}{7}}} + C \]

### Step 7: Back Substitute
Now substitute back \( t = x - 2 \):

\[ I = \frac{7}{5} \cdot \frac{(x - 2)^{\frac{8}{7}}}{(x + 3)^{\frac{8}{7}}} + C \]

### Final Answer
Thus, the final answer for the integral is:

\[ I = \frac{7}{5} \cdot \frac{(x - 2)^{\frac{8}{7}}}{(x + 3)^{\frac{8}{7}}} + C \]