`int((1-cos x)^(2//7))/((1+cosx)^(9//7))dx=`

Options

A. `(7)/(22)(tan^(11//7)x)+c`
B. `(7)/(11)tan^(11//7)((x)/(2))+c`
C. `(7)/(22)(1+cosx)^(11//7)+c`
D. `(7)/(11)(1-cosx)^(11//7)+c`

Answer

To solve the integral

\[ \int \frac{(1 - \cos x)^{\frac{2}{7}}}{(1 + \cos x)^{\frac{9}{7}}} \, dx, \]

we will use the substitution involving the tangent half-angle formula.

### Step 1: Use the half-angle identities
Recall the half-angle identities:
\[ \cos x = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)}. \]
Let \( t = \tan\left(\frac{x}{2}\right) \). Then, we have:
\[ 1 - \cos x = 1 - \frac{1 - t^2}{1 + t^2} = \frac{2t^2}{1 + t^2}, \]
and
\[ 1 + \cos x = 1 + \frac{1 - t^2}{1 + t^2} = \frac{2}{1 + t^2}. \]

### Step 2: Substitute into the integral
Now substituting these into the integral gives:
\[ \int \frac{\left(\frac{2t^2}{1 + t^2}\right)^{\frac{2}{7}}}{\left(\frac{2}{1 + t^2}\right)^{\frac{9}{7}}} \, dx. \]

### Step 3: Simplify the expression
This simplifies to:
\[ \int \frac{(2t^2)^{\frac{2}{7}}}{(1 + t^2)^{\frac{2}{7}}} \cdot \frac{(1 + t^2)^{\frac{9}{7}}}{2^{\frac{9}{7}}} \, dx. \]
This can be further simplified to:
\[ \int \frac{(2t^2)^{\frac{2}{7}}}{2^{\frac{9}{7}}} \cdot (1 + t^2)^{\frac{7}{7}} \, dx. \]

### Step 4: Change of variable for \(dx\)
We know that:
\[ dx = \frac{2}{1 + t^2} \, dt. \]
Substituting this into the integral gives:
\[ \int \frac{(2t^2)^{\frac{2}{7}}}{2^{\frac{9}{7}}} (1 + t^2) \cdot \frac{2}{1 + t^2} \, dt = \int \frac{(2t^2)^{\frac{2}{7}} \cdot 2}{2^{\frac{9}{7}}} \, dt. \]

### Step 5: Factor out constants
This simplifies to:
\[ \frac{2^{1 - \frac{9}{7}}}{1} \int t^{\frac{4}{7}} \, dt = 2^{-\frac{2}{7}} \int t^{\frac{4}{7}} \, dt. \]

### Step 6: Integrate
Now, integrating \( t^{\frac{4}{7}} \):
\[ \int t^{\frac{4}{7}} \, dt = \frac{t^{\frac{4}{7} + 1}}{\frac{4}{7} + 1} = \frac{t^{\frac{11}{7}}}{\frac{11}{7}} = \frac{7}{11} t^{\frac{11}{7}}. \]

### Step 7: Substitute back
Now substituting back \( t = \tan\left(\frac{x}{2}\right) \):
\[ \frac{2^{-\frac{2}{7}} \cdot 7}{11} \tan^{\frac{11}{7}}\left(\frac{x}{2}\right) + C. \]

### Final Answer
Thus, the final answer is:
\[ \frac{7}{11 \cdot 2^{\frac{2}{7}}} \tan^{\frac{11}{7}}\left(\frac{x}{2}\right) + C. \]