Give me some anti-differentiation problems.
Understand the Problem
The question is asking for examples of anti-differentiation problems, which are mathematical problems related to finding the antiderivatives of functions. This involves identifying functions whose derivatives yield given expressions, typically within the context of calculus.
Answer
Examples of anti-differentiation problems are: \( \int 2x \, dx = x^2 + C \), \( \int \sin(x) \, dx = -\cos(x) + C \), \( \int e^x \, dx = e^x + C \)
Answer for screen readers
Examples of anti-differentiation problems include:
- For ( f(x) = 2x ), ( \int 2x , dx = x^2 + C )
- For ( f(x) = \sin(x) ), ( \int \sin(x) , dx = -\cos(x) + C )
- For ( f(x) = e^x ), ( \int e^x , dx = e^x + C )
Steps to Solve
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Identify the Function First, we need to choose a function for which we want to find the antiderivative. For example, let's consider the function ( f(x) = 2x ).
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Apply the Power Rule for Antidifferentiation To find the antiderivative of ( f(x) ), we use the power rule for integration, which states that:
$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$
where ( n \neq -1 ) and ( C ) is the constant of integration.
In our case, ( n = 1 ):
$$ \int 2x , dx = 2 \cdot \frac{x^{1+1}}{1+1} + C = 2 \cdot \frac{x^2}{2} + C = x^2 + C $$
- Example with a Trigonometric Function Let's consider a second example: ( f(x) = \sin(x) ).
Using the known antiderivative:
$$ \int \sin(x) , dx = -\cos(x) + C $$
- Example with Exponential Function For an exponential function like ( f(x) = e^x ):
The antiderivative is:
$$ \int e^x , dx = e^x + C $$
- Summarizing the Results We can summarize the examples of antiderivative problems:
- From ( f(x) = 2x ), we find ( \int 2x , dx = x^2 + C )
- From ( f(x) = \sin(x) ), we find ( \int \sin(x) , dx = -\cos(x) + C )
- From ( f(x) = e^x ), we get ( \int e^x , dx = e^x + C )
Examples of anti-differentiation problems include:
- For ( f(x) = 2x ), ( \int 2x , dx = x^2 + C )
- For ( f(x) = \sin(x) ), ( \int \sin(x) , dx = -\cos(x) + C )
- For ( f(x) = e^x ), ( \int e^x , dx = e^x + C )
More Information
Antidifferentiation is a fundamental concept in calculus that allows us to find functions whose derivatives yield a given expression. Each example demonstrates a different type of function, showcasing the versatility of integration techniques.
Tips
- Forgetting to include the constant of integration ( C ) when finding antiderivatives.
- Misapplying the power rule, especially when dealing with negative or fractional exponents.
- Confusing the antiderivative with the derivative, leading to incorrect results.
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