Evaluate `int 2 sin(x)dx`.

Options

A. 0
B. `(1)/(3)`
C. `(2)/(3)`
D. 2

Answer

To evaluate the integral \( \int 2 \sin(x) \, dx \), we can follow these steps:

### Step 1: Set up the integral
Let \( I = \int 2 \sin(x) \, dx \).

### Step 2: Factor out the constant
Since 2 is a constant, we can factor it out of the integral:
\[ I = 2 \int \sin(x) \, dx \]

### Step 3: Integrate \( \sin(x) \)
We know that the integral of \( \sin(x) \) is:
\[ \int \sin(x) \, dx = -\cos(x) \]
So, substituting this into our equation gives:
\[ I = 2 \cdot (-\cos(x)) \]

### Step 4: Simplify the expression
Now, simplifying this, we have:
\[ I = -2 \cos(x) \]

### Step 5: Add the constant of integration
Since this is an indefinite integral, we must add the constant of integration \( C \):
\[ I = -2 \cos(x) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int 2 \sin(x) \, dx = -2 \cos(x) + C \]

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