`int (dx)/((x+a)^(8/7)(x-b)^(6/7))=`

Options

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

Answer

To solve the integral

\[ I = \int \frac{dx}{(x+a)^{8/7}(x-b)^{6/7}}, \]

we will follow a systematic approach.

### Step 1: Simplify the Integral
We start by multiplying and dividing the integrand by \((x+a)^{6/7}\):

\[ I = \int \frac{(x+a)^{6/7}}{(x+a)^{8/7}(x-b)^{6/7}} \, dx = \int \frac{dx}{(x+a)^{2}(x-b)^{6/7}}. \]

### Step 2: Rewrite the Integral
Now we can rewrite the integral as:

\[ I = \int \frac{dx}{(x+a)^{2}} \cdot \frac{1}{(x-b)^{6/7}}. \]

### Step 3: Substitution
Let us make the substitution:

\[ t = \frac{x-b}{x+a}. \]

From this substitution, we can express \(x\) in terms of \(t\):

\[ x - b = t(x + a) \implies x = \frac{b + ta}{1 - t}. \]

### Step 4: Differentiate to Find \(dx\)
Now we differentiate \(x\) with respect to \(t\):

\[ dx = \frac{d}{dt}\left(\frac{b + ta}{1 - t}\right) dt = \frac{a(1 - t) + (b + ta)}{(1 - t)^2} dt = \frac{a + b}{(1 - t)^2} dt. \]

### Step 5: Substitute Back into the Integral
Now substitute \(x\) and \(dx\) back into the integral:

\[ I = \int \frac{(1-t)^2}{(b + ta + a)^{2}} \cdot \frac{(1-t)^2}{(t)^{6/7}} \cdot \frac{a + b}{(1 - t)^2} dt. \]

### Step 6: Simplify the Integral
This simplifies to:

\[ I = (a + b) \int \frac{(1-t)^2}{(b + ta + a)^{2} t^{6/7}} dt. \]

### Step 7: Solve the Integral
Now we can solve this integral using standard integration techniques.

### Final Step: Back Substitute
After integrating, we will substitute back \(t = \frac{x-b}{x+a}\) to express the result in terms of \(x\).

### Conclusion
The final result will be:

\[ I = \frac{7}{a + b} \left(\frac{x-b}{x+a}\right)^{1/7} + C, \]

where \(C\) is the constant of integration.