Class CBSE Class 12 Mathematics Integrals Q #622
KNOWLEDGE BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
\(\int\frac{e^{x}}{\sqrt{4-e^{2x}}}dx\) is equal to:
(A) \(\frac{1}{2}\cos^{-1}(e^{x})+C\)
(B) \(\frac{1}{2}\sin^{-1}(e^{x})+C\)
(C) \(\frac{e^{x}}{2}+C\)
(D) \(\sin^{-1}(\frac{e^{x}}{2})+C\)
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Step-by-Step Solution

Let \(I = \int\frac{e^{x}}{\sqrt{4-e^{2x}}}dx\)

Substitute \(u = e^{x}\), then \(du = e^{x}dx\)

So, \(I = \int\frac{1}{\sqrt{4-u^{2}}}du\)

We know that \(\int\frac{1}{\sqrt{a^{2}-x^{2}}}dx = \sin^{-1}(\frac{x}{a})+C\)

Here, \(a = 2\), so \(I = \sin^{-1}(\frac{u}{2})+C\)

Substituting back \(u = e^{x}\), we get \(I = \sin^{-1}(\frac{e^{x}}{2})+C\)

Correct Answer: \(\sin^{-1}(\frac{e^{x}}{2})+C\)

AI Suggestion: Option D

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the standard integral formula after performing a suitable substitution.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure of substitution and then applying the standard integral formula to arrive at the solution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of integration techniques and standard integral formulas as covered in the textbook.

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