`int x cos x^(2)dx`

Answer

To solve the integral \( \int x \cos(x^2) \, dx \), we can use the method of substitution. Here’s a step-by-step solution:

### Step 1: Choose a substitution
Let \( t = x^2 \). Then, the differential \( dt \) can be found as follows:
\[ dt = 2x \, dx \quad \Rightarrow \quad \frac{1}{2} dt = x \, dx \]

### Step 2: Rewrite the integral
Substituting \( t = x^2 \) into the integral, we rewrite it:
\[ \int x \cos(x^2) \, dx = \int \cos(t) \cdot \frac{1}{2} dt \]
This simplifies to:
\[ \frac{1}{2} \int \cos(t) \, dt \]

### Step 3: Integrate
Now, we can integrate \( \cos(t) \):
\[ \int \cos(t) \, dt = \sin(t) \]
Thus, we have:
\[ \frac{1}{2} \sin(t) + C \]

### Step 4: Substitute back
Now, we substitute back \( t = x^2 \):
\[ \frac{1}{2} \sin(x^2) + C \]

### Final Answer
Therefore, the integral \( \int x \cos(x^2) \, dx \) is:
\[ \frac{1}{2} \sin(x^2) + C \]

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