Integral of 1/x from 1 to 2
Understand the Problem
The question is asking for the calculation of the definite integral of the function 1/x over the interval from 1 to 2. This involves finding the antiderivative of 1/x and evaluating it at the boundaries of the interval.
Answer
ln(2) ≈ 0.693
Answer for screen readers
The definite integral of 1/x from 1 to 2 is ln(2), which is approximately 0.693.
Steps to Solve
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Find the antiderivative
The antiderivative of $\frac{1}{x}$ is $\ln|x| + C$, where $C$ is the constant of integration.
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Apply the limits
Use the Fundamental Theorem of Calculus to evaluate the definite integral:
$$\int_{1}^{2} \frac{1}{x} dx = [\ln|x|]_{1}^{2} = \ln|2| - \ln|1|$$
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Simplify the result
Simplify the expression:
$\ln|2| - \ln|1| = \ln(2) - \ln(1) = \ln(2) - 0 = \ln(2)$
The natural logarithm of 2 is approximately 0.693.
The definite integral of 1/x from 1 to 2 is ln(2), which is approximately 0.693.
More Information
The natural logarithm of 2 (ln(2)) is an important mathematical constant. It appears in various mathematical and physical contexts, including information theory and thermodynamics. The value ln(2) is also equal to the area under the curve y = 1/x from x = 1 to x = 2.
Tips
A common mistake is forgetting to use the absolute value signs when writing the antiderivative of 1/x. Although it doesn't affect the result in this case (since the interval is positive), it's important to include |x| for the general antiderivative: ln|x| + C.
