The integral `I=int[xe^(x^(2))(sinx^(2)+cosx^(2))]dx =f(x)+c`, (where, c is the constant of integration). Then, f(x) can be

Options

A. `e^(x)sin(x^(2))`
B. `e^(x^(2))sin(x)`
C. `e^(x^(2))((x^(2))/(2))`
D. `(1)/(2)e^(x^(2))sin(x^(2))`

Answer

To solve the integral

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

we can follow these steps:

### Step 1: Substitution
Let \( t = x^2 \). Then, the differential \( dt = 2x \, dx \) implies that \( x \, dx = \frac{dt}{2} \).

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral, we have:

\[ I = \int x e^{x^2} (\sin x^2 + \cos x^2) \, dx = \int e^t (\sin t + \cos t) \frac{dt}{2}. \]

### Step 3: Factor Out Constants
We can factor out the constant \( \frac{1}{2} \):

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

### Step 4: Integration by Parts
Next, we can use integration by parts. We know the formula:

\[ \int e^x f(x) \, dx = e^x f(x) - \int e^x f'(x) \, dx. \]

Let \( f(t) = \sin t + \cos t \). Then, \( f'(t) = \cos t - \sin t \).

### Step 5: Apply Integration by Parts
Applying integration by parts, we have:

\[ \int e^t (\sin t + \cos t) \, dt = e^t (\sin t + \cos t) - \int e^t (\cos t - \sin t) \, dt. \]

### Step 6: Simplify the Integral
Now we need to evaluate the integral \( \int e^t (\cos t - \sin t) \, dt \). We can apply integration by parts again for this integral.

### Step 7: Combine Results
After performing the integration by parts twice, we will combine the results. The final result will be in terms of \( e^t \), which we will substitute back to \( x \) using \( t = x^2 \).

### Step 8: Final Expression
The integral \( I \) will yield a function \( f(x) \) plus a constant of integration \( c \):

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

Thus, we can conclude that:

\[ f(x) = \frac{1}{2} e^{x^2} (\sin x^2 + \cos x^2). \]