The value of integral
`int1/[(x-3)^(3)(x+2)^(5)]^(1//4)dx` is

Options

A. `4/3(x-1)/(x+2)^(1//4)` dx is equal to
B. `sqrt(x)sqrt(1-x)-2sqrt(1-x)+cos^(-1)(sqrt(x))+C`
C. `sqrt(x)sqrt(1-x)+2sqrt(1-x)+cos^(-)(sqrt(x))+C`
D. `sqrt(x)sqrt(1-x)-2sqrt(1-x)-cos^(-1)sqrt(x)+C`

Explanation

To solve the integral

\[ \int \frac{1}{\left[(x-3)^3 (x+2)^5\right]^{\frac{1}{4}}} \, dx, \]

we can start by simplifying the expression inside the integral.

### Step 1: Rewrite the Integral
We can rewrite the integral as:

\[ \int \frac{1}{(x-3)^{\frac{3}{4}} (x+2}^{\frac{5}{4}}) \, dx. \]

### Step 2: Multiply and Divide by \( (x+2)^{\frac{3}{4}} \)
Next, we multiply and divide by \( (x+2)^{\frac{3}{4}} \):

\[ \int \frac{(x+2)^{\frac{3}{4}}}{(x-3)^{\frac{3}{4}} (x+2)^{\frac{3}{4}} (x+2)^{\frac{5}{4}}} \, dx. \]

This simplifies to:

\[ \int \frac{(x+2)^{\frac{3}{4}}}{(x-3)^{\frac{3}{4}} (x+2)^{2}} \, dx. \]

### Step 3: Combine the Powers
Now, we can combine the powers of \( (x+2) \):

\[ \int \frac{(x+2)^{\frac{3}{4}}}{(x-3)^{\frac{3}{4}} (x+2)^{2}} \, dx = \int \frac{1}{(x-3)^{\frac{3}{4}} (x+2)^{\frac{5}{4}}} \, dx. \]

### Step 4: Substitution
Let us make the substitution:

\[ y = \frac{x-3}{x+2}. \]

Differentiating both sides gives:

\[ dy = \frac{(x+2)(1) - (x-3)(1)}{(x+2)^2} \, dx = \frac{5}{(x+2)^2} \, dx. \]

Thus, we have:

\[ dx = \frac{(x+2)^2}{5} \, dy. \]

### Step 5: Substitute Back into the Integral
Substituting \( dx \) into the integral gives:

\[ \int \frac{1}{y^{\frac{3}{4}}} \cdot \frac{(x+2)^2}{5} \, dy. \]

### Step 6: Simplifying the Integral
Now, we can simplify the integral:

\[ \frac{1}{5} \int y^{-\frac{3}{4}} (x+2)^2 \, dy. \]

### Step 7: Integrate
Integrating \( y^{-\frac{3}{4}} \):

\[ \frac{1}{5} \cdot \left( \frac{y^{\frac{1}{4}}}{\frac{1}{4}} \right) + C = \frac{4}{5} y^{\frac{1}{4}} + C. \]

### Step 8: Substitute Back for \( y \)
Now substituting back for \( y \):

\[ \frac{4}{5} \left( \frac{x-3}{x+2} \right)^{\frac{1}{4}} + C. \]

### Final Result
Thus, the final answer for the integral is:

\[ \frac{4}{5} \left( \frac{x-3}{x+2} \right)^{\frac{1}{4}} + C. \]

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