Question:
The value of $\int_{1}^{e}\frac{1}{x}dx$ is equal to
The value of $\int_{1}^{e}\frac{1}{x}dx$ is equal to
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Integration Tip: Remember that in calculus contexts, $\log(x)$ overwhelmingly refers to the natural logarithm $\ln(x)$ with base $e$.
Updated On: Apr 30, 2026
- e
- 1
- $e^{2}$
- $e^{2}-e$
- 0
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The Correct Option is B
Solution and Explanation
Concept:
The integral of $\frac{1}{x}$ with respect to $x$ is one of the foundational calculus formulas: $\int \frac{1}{x} dx = \log|x| + C$ (using the natural logarithm). Once the antiderivative is found, use the Fundamental Theorem of Calculus to evaluate the bounds.
Step 1: Find the antiderivative.
Integrate the function $\frac{1}{x}$: $$\int \frac{1}{x} dx = \log|x|$$
Step 2: Set up the evaluation limits.
Apply the limits of integration from $1$ to $e$: $$[\log|x|]_{1}^{e}$$
Step 3: Substitute the upper limit.
Plug the upper limit $e$ into the antiderivative: $$\log|e| = 1$$ *(Note: The natural logarithm of Euler's number $e$ is exactly 1).*
Step 4: Substitute the lower limit.
Plug the lower limit $1$ into the antiderivative: $$\log|1| = 0$$
Step 5: Subtract to find the final value.
Subtract the lower limit evaluation from the upper limit evaluation: $$1 - 0 = 1$$ Hence the correct answer is (B) 1.
The integral of $\frac{1}{x}$ with respect to $x$ is one of the foundational calculus formulas: $\int \frac{1}{x} dx = \log|x| + C$ (using the natural logarithm). Once the antiderivative is found, use the Fundamental Theorem of Calculus to evaluate the bounds.
Step 1: Find the antiderivative.
Integrate the function $\frac{1}{x}$: $$\int \frac{1}{x} dx = \log|x|$$
Step 2: Set up the evaluation limits.
Apply the limits of integration from $1$ to $e$: $$[\log|x|]_{1}^{e}$$
Step 3: Substitute the upper limit.
Plug the upper limit $e$ into the antiderivative: $$\log|e| = 1$$ *(Note: The natural logarithm of Euler's number $e$ is exactly 1).*
Step 4: Substitute the lower limit.
Plug the lower limit $1$ into the antiderivative: $$\log|1| = 0$$
Step 5: Subtract to find the final value.
Subtract the lower limit evaluation from the upper limit evaluation: $$1 - 0 = 1$$ Hence the correct answer is (B) 1.
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