Class CBSE Class 12 Mathematics Definite Integrals Q #634
KNOWLEDGE BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
\(\int_{0}^{\pi/2}\cos x\cdot e^{\sin x}dx\) is equal to:
(A) 0
(B) \(1-e\)
(C) \(e-1\)
(D) e
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AI Tutor Explanation

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Step-by-Step Solution

Let \(I = \int_{0}^{\pi/2}\cos x\cdot e^{\sin x}dx\)

Let \(u = \sin x\). Then, \(\frac{du}{dx} = \cos x\), so \(du = \cos x \, dx\).

When \(x = 0\), \(u = \sin(0) = 0\).

When \(x = \pi/2\), \(u = \sin(\pi/2) = 1\).

So, the integral becomes \(I = \int_{0}^{1} e^u du\).

The integral of \(e^u\) is \(e^u\), so \(I = [e^u]_{0}^{1}\).

Evaluating the limits, \(I = e^1 - e^0 = e - 1\).

Correct Answer: e-1

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the knowledge of integration techniques, specifically substitution, to solve the definite integral.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to execute a specific procedure (integration using substitution) 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 ability to apply integration techniques learned from the textbook.

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