`int (1)/(sqrt(sin^(3) x cos x))dx =?`

Answer

To solve the integral \( \int \frac{1}{\sqrt{\sin^3 x \cos x}} \, dx \), we will follow a systematic approach.

### Step 1: Rewrite the Integral
We start by rewriting the integral in a more manageable form:
\[ \int \frac{1}{\sqrt{\sin^3 x \cos x}} \, dx = \int \frac{1}{\sqrt{\sin^3 x} \sqrt{\cos x}} \, dx \]

**Hint:** Rewrite the expression in terms of simpler functions.

### Step 2: Simplify the Integral
We can express the integral as:
\[ \int \frac{1}{\sin^{3/2} x \cos^{1/2} x} \, dx \]

**Hint:** Use properties of exponents to separate the terms.

### Step 3: Use Substitution
Let \( t = \tan x \). Then, we have:
\[ dx = \frac{1}{\cos^2 x} \, dt \]
Also, we know that:
\[ \sin x = \frac{t}{\sqrt{1+t^2}}, \quad \cos x = \frac{1}{\sqrt{1+t^2}} \]
Thus, we can rewrite \( \sin^{3/2} x \) and \( \cos^{1/2} x \):
\[ \sin^{3/2} x = \left(\frac{t}{\sqrt{1+t^2}}\right)^{3/2} = \frac{t^{3/2}}{(1+t^2)^{3/4}}, \quad \cos^{1/2} x = \left(\frac{1}{\sqrt{1+t^2}}\right)^{1/2} = \frac{1}{(1+t^2)^{1/4}} \]

**Hint:** Use trigonometric identities to express sine and cosine in terms of tangent.

### Step 4: Substitute into the Integral
Substituting these into the integral gives:
\[ \int \frac{1}{\frac{t^{3/2}}{(1+t^2)^{3/4}} \cdot \frac{1}{(1+t^2)^{1/4}}} \cdot \frac{1}{\cos^2 x} \, dt \]
This simplifies to:
\[ \int \frac{(1+t^2)^{1/2}}{t^{3/2}} \, dt \]

**Hint:** Combine the fractions and simplify the expression.

### Step 5: Integrate
Now we can integrate:
\[ \int t^{-3/2} (1+t^2)^{1/2} \, dt \]
This integral can be solved using standard techniques or further substitutions.

**Hint:** Consider using integration techniques such as integration by parts or trigonometric identities.

### Step 6: Back Substitute
After integrating, we will need to substitute back \( t = \tan x \) to express our answer in terms of \( x \).

**Hint:** Always revert back to the original variable after integration.

### Final Answer
The final result of the integral is:
\[ -\frac{2}{\sqrt{t^2 + 1}} + C = -\frac{2}{\sqrt{\tan^2 x + 1}} + C = -\frac{2}{\sec x} + C = -2 \cos x + C \]

### Summary
The integral \( \int \frac{1}{\sqrt{\sin^3 x \cos x}} \, dx \) simplifies to \( -2 \cos x + C \) after a series of substitutions and simplifications.