Evaluate: `int(sqrt((cosx)/(x))-sqrt((x)/(cosx))sinx)dx` equals

Options

A. `-sqrt(xcosx)+C`
B. `sqrt(xsinx)+C`
C. `2sqrt(xcos x)+C`
D. `C-2sqrt(xcosx)`

Explanation

To evaluate the integral

\[ I = \int \left( \sqrt{\frac{\cos x}{x}} - \sqrt{\frac{x}{\cos x}} \sin x \right) dx, \]

we will follow these steps:

### Step 1: Combine the terms under a common denominator

We can rewrite the integral as:

\[ I = \int \left( \frac{\sqrt{\cos x} - \sqrt{x} \sin x \sqrt{\cos x}}{\sqrt{x \cos x}} \right) dx. \]

### Step 2: Factor the numerator

The numerator can be factored as:

\[ \sqrt{\cos x} (1 - \sqrt{x} \sin x). \]

Thus, we have:

\[ I = \int \frac{\sqrt{\cos x} (1 - \sqrt{x} \sin x)}{\sqrt{x \cos x}} dx. \]

### Step 3: Simplify the integral

This simplifies to:

\[ I = \int \frac{1 - \sqrt{x} \sin x}{\sqrt{x}} dx. \]

### Step 4: Substitute for easier integration

Let \( t = x \cos x \). Then, we differentiate \( t \):

\[ dt = (\cos x - x \sin x) dx. \]

Rearranging gives us:

\[ dx = \frac{dt}{\cos x - x \sin x}. \]

### Step 5: Substitute into the integral

Substituting \( t \) into our integral, we have:

\[ I = \int \frac{1}{\sqrt{t}} dt. \]

### Step 6: Integrate

The integral of \( t^{-1/2} \) is:

\[ \int t^{-1/2} dt = 2\sqrt{t} + C. \]

### Step 7: Substitute back for \( t \)

Now, substituting back for \( t \):

\[ I = 2\sqrt{x \cos x} + C. \]

### Final Result

Thus, the evaluated integral is:

\[ I = 2\sqrt{x \cos x} + C. \]

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