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Let $\beta$ be the area of the region enclosed between the curves $C_1, C_2$, and the lines \[ x = \alpha_1 \quad \text{and} \quad x = \alpha_4. \] Then the value of \[ -\frac{1}{\pi}\ln\!\left(\beta - 2e^{-\pi/2}\right) \] is:

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The integral $\int e^{-x}(\sin x + \cos x) dx$ is a "perfect derivative" of $-e^{-x}\sin x$. Recognizing such patterns saves significant time during complex integration steps in competitive exams.

Updated On: May 22, 2026

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Correct Answer: 2

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Let \(\beta\) be the area of the region enclosed between the curves \(C_1, C_2\), and the lines \[ x = \alpha_1 \quad \text{and} \quad x = \alpha_4. \] Then the value of \[ -\frac{1}{\pi}\ln\!\left(\beta - 2e^{-\pi/2}\right) \] is:

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Correct Answer: 2

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