The value of `int(cos^4xdx)/(sin^3x(sin^5x+cos^5x)^(3//5))=-(1)/(2)(1+g(x))^(2//5)+C` where g (x) is :

Options

A. `tan^(5)x`
B. `cot^(5)x`
C. `sin^(5)x`
D. `cos^(5)x`

Answer

To solve the integral

\[ \int \frac{\cos^4 x \, dx}{\sin^3 x \left( \sin^5 x + \cos^5 x \right)^{3/5}} = -\frac{1}{2} \left( 1 + g(x) \right)^{2/5} + C \]

we need to determine the function \( g(x) \).

### Step-by-Step Solution:

1. **Identify the Structure of the Integral**:
The integral involves powers of sine and cosine, and we need to simplify the expression in the denominator.

2. **Substitution**:
Let's consider the substitution \( g(x) = \cot^5 x \). This is a reasonable assumption because it relates to the powers of sine and cosine.

3. **Differentiate \( g(x) \)**:
We differentiate \( g(x) \):
\[ g'(x) = 5 \cot^4 x (-\csc^2 x) = -5 \cot^4 x \csc^2 x \]

4. **Substituting \( g(x) \) into the Integral**:
We substitute \( g(x) \) back into the equation:
\[ 1 + g(x) = 1 + \cot^5 x \]

5. **Rewrite the Integral**:
The integral can be rewritten using our substitution:
\[ \int \frac{\cos^4 x \, dx}{\sin^3 x \left( \sin^5 x + \cos^5 x \right)^{3/5}} = -\frac{1}{2} \left( 1 + \cot^5 x \right)^{2/5} + C \]

6. **Verify the Result**:
To ensure that our choice of \( g(x) \) is correct, we can differentiate the right-hand side and check if it matches the left-hand side of the integral.

### Final Result:
Thus, we find that

\[ g(x) = \cot^5 x \]