The value of `int(dx)/(cos^3x sqrtsin2x)` is equal to

Options

A. `sqrt(2)(cossqrt(cosx)+1/2tan^(5//2)x)+C`
B. `sqrt(2)(sqrt(tanx)+1/5tan^(5//2)x)+C`
C. `sqrt(2)(sqrt(tanx)-1/5tan^(5//2)x)+C`
D. `sqrt(2)(sqrt(cosx)-1/5tan^(5//2)x)+C`

Answer

To solve the integral

\[ I = \int \frac{dx}{\cos^3 x \sqrt{\sin 2x}}, \]

we will follow these steps:

### Step 1: Rewrite \(\sin 2x\)
We know that

\[ \sin 2x = 2 \sin x \cos x. \]

Thus, we can rewrite the integral as:

\[ I = \int \frac{dx}{\cos^3 x \sqrt{2 \sin x \cos x}}. \]

### Step 2: Simplify the square root
We can express the square root as:

\[ \sqrt{2 \sin x \cos x} = \sqrt{2} \sqrt{\sin x} \sqrt{\cos x}. \]

So the integral becomes:

\[ I = \int \frac{dx}{\cos^3 x \sqrt{2} \sqrt{\sin x} \sqrt{\cos x}} = \frac{1}{\sqrt{2}} \int \frac{dx}{\cos^{3.5} x \sqrt{\sin x}}. \]

### Step 3: Rewrite the integral
Now, we can rewrite the integral in terms of powers of sine and cosine:

\[ I = \frac{1}{\sqrt{2}} \int \frac{dx}{\cos^{7/2} x \sin^{1/2} x}. \]

### Step 4: Multiply and divide by \(\cos^{1/2} x\)
To facilitate integration, we multiply and divide by \(\cos^{1/2} x\):

\[ I = \frac{1}{\sqrt{2}} \int \frac{\cos^{1/2} x}{\cos^{4} x \sin^{1/2} x} \, dx = \frac{1}{\sqrt{2}} \int \frac{1}{\sin^{1/2} x} \cdot \frac{1}{\cos^{4} x} \, dx. \]

### Step 5: Use the substitution \(t = \tan x\)
Let \(t = \tan x\), then \(dx = \sec^2 x \, dt\) and \(\sin x = \frac{t}{\sqrt{1+t^2}}\), \(\cos x = \frac{1}{\sqrt{1+t^2}}\).

Substituting these into the integral gives:

\[ I = \frac{1}{\sqrt{2}} \int \frac{\sec^2 x}{\left(\frac{1}{\sqrt{1+t^2}}\right)^{4} \left(\frac{t}{\sqrt{1+t^2}}\right)^{1/2}} \, dt. \]

### Step 6: Simplify the integral
This simplifies to:

\[ I = \frac{1}{\sqrt{2}} \int \frac{(1+t^2)^2}{t^{1/2}} \, dt. \]

### Step 7: Integrate
Now we can integrate term by term. The integral can be computed using standard integration techniques.

### Step 8: Substitute back
After integrating, we substitute back \(t = \tan x\) to express the result in terms of \(x\).

### Final Result
The final result will be:

\[ I = \sqrt{2} \left(\sqrt{\tan x} + \frac{1}{5} \tan^{5/2} x\right) + C. \]