`int 2 sin (x)dx` is equal to :

Options

A. `-2 cos x +C`
B. `2 cos x +C`
C. `-2 cos x`
D. `2 cos x`

Answer

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

### Step 1: Factor out the constant
The integral can be simplified by factoring out the constant 2:
\[ \int 2 \sin(x) \, dx = 2 \int \sin(x) \, dx \]

### Step 2: Integrate \( \sin(x) \)
Now, we need to find the integral of \( \sin(x) \):
\[ \int \sin(x) \, dx = -\cos(x) + C \]
where \( C \) is the constant of integration.

### Step 3: Multiply by the constant
Now, we multiply the result by the constant we factored out in Step 1:
\[ 2 \int \sin(x) \, dx = 2 \left( -\cos(x) + C \right) = -2\cos(x) + 2C \]

### Step 4: Simplify the expression
Since \( 2C \) is still a constant, we can denote it as just \( C \) (as it represents an arbitrary constant):
\[ \int 2 \sin(x) \, dx = -2\cos(x) + C \]

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

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