integral of cos(2t)
Understand the Problem
The question is asking for the integral of the function cos(2t) with respect to t. To solve it, we will apply the rules of integration, specifically for trigonometric functions.
Answer
The integral of $ \cos(2t) $ with respect to $ t $ is $ \frac{1}{2} \sin(2t) + C $.
Answer for screen readers
The integral of $ \cos(2t) $ with respect to $ t $ is $ \frac{1}{2} \sin(2t) + C $.
Steps to Solve
- Identify the function to integrate
We need to integrate the function $ \cos(2t) $ with respect to $ t $.
- Use the integration rule for cosine
The integral of $ \cos(kx) $ with respect to $ x $ is given by:
$$ \int \cos(kx) , dx = \frac{1}{k} \sin(kx) + C $$
where $ C $ is the constant of integration. In our case, ( k = 2 ).
- Apply the formula
Substituting ( k = 2 ) into the formula gives us:
$$ \int \cos(2t) , dt = \frac{1}{2} \sin(2t) + C $$
- Write the final result
Thus, the integral of $ \cos(2t) $ with respect to $ t $ is:
$$ \frac{1}{2} \sin(2t) + C $$
The integral of $ \cos(2t) $ with respect to $ t $ is $ \frac{1}{2} \sin(2t) + C $.
More Information
The integration of cosine functions typically results in a sine function, and the coefficient in front of the argument of the cosine (in this case, 2) becomes a divisor. Integrating trigonometric functions is fundamental in calculus and often appears in physics and engineering.
Tips
- Forgetting to include the constant of integration $ C $ after performing the integration.
- Confusing the integral of cosine with sine, which could lead to incorrect results.
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