The value of the integral `int(xsin x^(2)e^(secx^(2)))/(cos^(2)x^(2))dx` , is

Options

A. `(1)/(2)e^(secx^(2))+C`
B. `(1)/(2)e^(sinx^(2))+C`
C. `(1)/(2)sinx^(2)e^(cos^(2)x^(2))+C`
D. none of these

Answer

To solve the integral

\[ I = \int \frac{x \sin(x^2) e^{\sec(x^2)}}{\cos^2(x^2)} \, dx, \]

we will follow these steps:

### Step 1: Simplify the Integral
We can rewrite the integral using the identity for \( \sec(x^2) \) and \( \cos^2(x^2) \):

\[ I = \int x \sin(x^2) e^{\sec(x^2)} \sec^2(x^2) \, dx. \]

### Step 2: Substitution
Let’s make the substitution:

\[ t = \sec(x^2). \]

Then, we differentiate \( t \):

\[ dt = \sec(x^2) \tan(x^2) \cdot 2x \, dx. \]

This gives us:

\[ \frac{dt}{2} = x \sec(x^2) \tan(x^2) \, dx. \]

### Step 3: Rewrite the Integral
Now we can rewrite the integral in terms of \( t \):

\[ I = \int e^t \cdot \frac{dt}{2}. \]

### Step 4: Integrate
Now, we can integrate:

\[ I = \frac{1}{2} \int e^t \, dt = \frac{1}{2} e^t + C. \]

### Step 5: Substitute Back
Now, substituting back for \( t \):

\[ I = \frac{1}{2} e^{\sec(x^2)} + C. \]

### Final Result
Thus, the value of the integral is:

\[ I = \frac{1}{2} e^{\sec(x^2)} + C. \]