`int (dx)/((x + alpha)^(8//7) (x - beta)^(6//7)) = `

Options

A. `6/(alpha + beta)((x - beta)/(x + alpha))^(1//6)`
B. `6/(alpha + beta) ((x + alpha)/(x - beta))^(1//6)`
C. `7/(alpha + beta) ((x + alpha)/(x - beta))^(1//7)`
D. `7/(alpha + beta) ((x - beta)/(x + alpha))^(1//7)`

Answer

To solve the integral

\[ \int \frac{dx}{(x + \alpha)^{\frac{8}{7}} (x - \beta)^{\frac{6}{7}}} \]

we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral in a more manageable form. We can express the integral as:

\[ \int \frac{1}{(x + \alpha)^{\frac{8}{7}} (x - \beta)^{\frac{6}{7}}} \, dx \]

### Step 2: Break Down the Powers
Next, we can separate the powers in the denominator:

\[ = \int \frac{1}{(x + \alpha)^{\frac{8}{7}}} \cdot \frac{1}{(x - \beta)^{\frac{6}{7}}} \, dx \]

### Step 3: Use Substitution
We will use the substitution:

\[ t = \frac{x - \beta}{x + \alpha} \]

Now we need to find \(dx\) in terms of \(dt\). First, we differentiate \(t\):

\[ dt = \frac{(x + \alpha)(1) - (x - \beta)(1)}{(x + \alpha)^2} \, dx = \frac{(x + \alpha) - (x - \beta)}{(x + \alpha)^2} \, dx \]

This simplifies to:

\[ dt = \frac{\beta + \alpha}{(x + \alpha)^2} \, dx \]

Thus,

\[ dx = \frac{(x + \alpha)^2}{\beta + \alpha} \, dt \]

### Step 4: Substitute Back into the Integral
Now substitute \(dx\) and \(t\) back into the integral:

\[ \int \frac{(x + \alpha)^2}{\beta + \alpha} \cdot \frac{1}{(x + \alpha)^{\frac{8}{7}}} \cdot \frac{1}{(x - \beta)^{\frac{6}{7}}} \, dt \]

This simplifies to:

\[ \frac{1}{\beta + \alpha} \int \frac{(x + \alpha)^{2 - \frac{8}{7}}}{(x - \beta)^{\frac{6}{7}}} \, dt \]

### Step 5: Simplify the Powers
Now we simplify the powers:

\[ = \frac{1}{\beta + \alpha} \int \frac{(x + \alpha)^{\frac{6}{7}}}{(x - \beta)^{\frac{6}{7}}} \, dt \]

### Step 6: Perform the Integral
Now we can perform the integral:

\[ = \frac{1}{\beta + \alpha} \cdot \frac{7}{1} \cdot t^{\frac{1}{7}} + C \]

### Step 7: Substitute Back for \(t\)
Finally, substitute back for \(t\):

\[ = \frac{7}{\beta + \alpha} \left(\frac{x - \beta}{x + \alpha}\right)^{\frac{1}{7}} + C \]

### Final Answer
Thus, the final result of the integral is:

\[ \frac{7}{\beta + \alpha} \left(\frac{x - \beta}{x + \alpha}\right)^{\frac{1}{7}} + C \]