integral of 2sin(x)
Understand the Problem
The question is asking to find the integral of the function 2sin(x). This involves using basic integration rules for trigonometric functions.
Answer
The integral of $2\sin(x)$ is $-2\cos(x) + C$.
Answer for screen readers
The integral of the function $2\sin(x)$ is
$$ -2\cos(x) + C $$
Steps to Solve
- Identify the Function to Integrate
The function we need to integrate is $2\sin(x)$.
- Apply the Constant Multiple Rule
When integrating, we can factor out constants from the integral:
$$ \int 2\sin(x) , dx = 2 \int \sin(x) , dx $$
- Integrate the Sine Function
The integral of $\sin(x)$ is known:
$$ \int \sin(x) , dx = -\cos(x) + C $$
- Substitute Back into the Equation
Now, we substitute the result back into our equation:
$$ 2 \int \sin(x) , dx = 2(-\cos(x) + C) $$
- Simplify the Expression
Distributing the constant:
$$ -2\cos(x) + 2C $$
Since $C$ is an arbitrary constant, we can simply denote it as a single constant $C$:
$$ -2\cos(x) + C $$
The integral of the function $2\sin(x)$ is
$$ -2\cos(x) + C $$
More Information
This result shows how trigonometric integrals can be effectively solved by recognizing integration rules. The inclusion of the constant $C$ represents all possible antiderivatives of the function.
Tips
- Neglecting the Constant: Forgetting to include the constant of integration $C$ can lead to an incomplete answer. Always remember to add it when performing indefinite integrals.
- Misremembering the Integral of Sine: Confusing the integral of $\sin(x)$ with another function. The correct integral is $-\cos(x)$.
