Evaluate `int x sin(4x^(2)+7 ) dx`

Answer

To evaluate the integral \( \int x \sin(4x^2 + 7) \, dx \), we will use substitution. Here are the steps:

### Step 1: Substitution
Let \( t = 4x^2 + 7 \).

### Step 2: Differentiate
Now, differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = 8x \implies dt = 8x \, dx \implies x \, dx = \frac{dt}{8} \]

### Step 3: Rewrite the Integral
Substituting \( t \) and \( x \, dx \) into the integral, we have:
\[ \int x \sin(4x^2 + 7) \, dx = \int \sin(t) \cdot \frac{dt}{8} \]

### Step 4: Factor Out the Constant
We can factor out the constant \( \frac{1}{8} \):
\[ = \frac{1}{8} \int \sin(t) \, dt \]

### Step 5: Integrate
The integral of \( \sin(t) \) is:
\[ \int \sin(t) \, dt = -\cos(t) + C \]
Thus,
\[ \frac{1}{8} \int \sin(t) \, dt = \frac{1}{8} (-\cos(t) + C) = -\frac{1}{8} \cos(t) + \frac{C}{8} \]

### Step 6: Substitute Back
Now substitute back \( t = 4x^2 + 7 \):
\[ = -\frac{1}{8} \cos(4x^2 + 7) + C \]

### Final Answer
The final result is:
\[ \int x \sin(4x^2 + 7) \, dx = -\frac{1}{8} \cos(4x^2 + 7) + C \]