`int cos^(-3//7)x sin^(-11//7)x dx` is equal to

Options

A. `log|sin^(4/7)x|+C`
B. `(4)/(7)tan^(4//7)x+C`
C. `-(7)/(4)tan^(-4//7)x+C`
D. `log|cos^(3//7)x|+C`

Explanation

To solve the integral \( \int \cos^{-\frac{3}{7}}(x) \sin^{-\frac{11}{7}}(x) \, dx \), we will follow a systematic approach.

### Step 1: Rewrite the Integral
We can rewrite the integral in a more manageable form. We will express the integral as:

\[ \int \frac{1}{\cos^{\frac{3}{7}}(x) \sin^{\frac{11}{7}}(x)} \, dx \]

### Step 2: Multiply and Divide by a Suitable Factor
To simplify the integral, we can multiply and divide by \( \cos^{-\frac{11}{7}}(x) \):

\[ \int \frac{\cos^{-\frac{11}{7}}(x)}{\cos^{-\frac{3}{7}}(x) \sin^{-\frac{11}{7}}(x) \cos^{-\frac{11}{7}}(x)} \, dx \]

This gives us:

\[ \int \frac{\cos^{-\frac{11}{7}}(x)}{\sin^{-\frac{11}{7}}(x)} \cdot \frac{1}{\cos^{-\frac{3}{7}}(x)} \, dx \]

### Step 3: Use Trigonometric Identities
Now we can use the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) to rewrite the integral:

\[ \int \tan^{\frac{11}{7}}(x) \cdot \sec^{\frac{3}{7}}(x) \, dx \]

### Step 4: Make a Substitution
Let’s make the substitution \( u = \tan(x) \), which gives us \( du = \sec^2(x) \, dx \) or \( dx = \frac{du}{\sec^2(x)} \).

Since \( \sec(x) = \sqrt{1 + \tan^2(x)} = \sqrt{1 + u^2} \), we can rewrite the integral in terms of \( u \):

\[ \int u^{\frac{11}{7}} \cdot (1 + u^2)^{\frac{3}{14}} \, du \]

### Step 5: Solve the Integral
This integral can now be solved using integration techniques such as integration by parts or substitution depending on the complexity.

### Step 6: Back Substitute
After solving the integral in terms of \( u \), we will back substitute \( u = \tan(x) \) to express our final answer in terms of \( x \).

### Final Answer
The final answer will be in the form of:

\[ \text{Result} + C \]

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

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