Class CBSE Class 12 Mathematics Definite Integrals Q #1709
KNOWLEDGE BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
The value of $\int_{-5}^{-1}\frac{1}{x}dx$ is equal to:
(A) $-\log 5$
(B) $x^{6}$
(C) $\log(-5)$
(D) $x^{-6}$
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Correct Answer: A

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Detailed Solution

Step 1: Identify the Integral

We need to evaluate the definite integral of the function $f(x) = \frac{1}{x}$ over the interval $[-5, -1]$. The integral is given by: $$I = \int_{-5}^{-1} \frac{1}{x} dx$$

Step 2: Apply the Fundamental Theorem of Calculus

The antiderivative of $\frac{1}{x}$ is $\ln|x|$. Applying the limits from $-5$ to $-1$: $$I = [\ln|x|]_{-5}^{-1}$$

Step 3: Substitute the Limits

Substitute the upper and lower limits into the antiderivative: $$I = \ln|-1| - \ln|-5|$$ $$I = \ln(1) - \ln(5)$$

Step 4: Simplify the Result

Since $\ln(1) = 0$, the expression simplifies to: $$I = 0 - \ln(5) = -\ln(5)$$

Final Answer: -log 5

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the Fundamental Theorem of Calculus and the properties of logarithmic functions to a specific definite integral.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the execution of a standard algorithmic process (integration) and the handling of absolute values within logarithmic functions.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It tests the foundational understanding of definite integrals as prescribed in the Calculus unit of the NCERT curriculum.
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