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

Explanation

To solve the integral

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

we will follow these steps:

### Step 1: Substitution
Let \( t = x^2 \). Then, we have:

\[ dt = 2x \, dx \quad \Rightarrow \quad dx = \frac{dt}{2x}. \]

### Step 2: Rewrite the integral
Now, substituting \( x^2 = t \) into the integral, we get:

\[ \int \frac{x \sin(t) e^{\sec(t)}}{\cos^2(t)} \cdot \frac{dt}{2x}. \]

The \( x \) in the numerator and denominator cancels out:

\[ = \frac{1}{2} \int \frac{\sin(t) e^{\sec(t)}}{\cos^2(t)} \, dt. \]

### Step 3: Simplify the integral
Recall that \( \sec(t) = \frac{1}{\cos(t)} \), so we can rewrite \( e^{\sec(t)} \) as \( e^{1/\cos(t)} \). Thus, the integral becomes:

\[ = \frac{1}{2} \int \sin(t) e^{\frac{1}{\cos(t)}} \sec^2(t) \, dt. \]

### Step 4: Further substitution
Now, we can use another substitution. Let \( u = \sec(t) \), then:

\[ \frac{du}{dt} = \sec(t) \tan(t) \quad \Rightarrow \quad dt = \frac{du}{\sec(t) \tan(t)}. \]

### Step 5: Substitute and simplify
Substituting \( u = \sec(t) \) into our integral, we have:

\[ = \frac{1}{2} \int \sin(t) e^u \cdot \frac{du}{\sec(t) \tan(t)}. \]

### Step 6: Integrate
This integral can be simplified further, but it will require knowledge of integration techniques. The integral of \( e^u \) is straightforward:

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

### Step 7: Back substitution
Finally, substitute back \( t = x^2 \):

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

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

\[ \int \frac{x \sin(x^2) e^{\sec(x^2)}}{\cos^2(x^2)} \, dx = \frac{1}{2} e^{\sec(x^2)} + C. \]

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