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Need solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Fill in the blanks question 3

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Answer: e-1

Hint: You must know about \frac{d}{d x}(\sin x) \text { and } \int e^{x} d x

Given:\int_{0}^{\frac{\pi}{2}} e^{\sin x} \cos x \; dx


I=\int_{0}^{\frac{\pi}{2}} e^{\sin x} \cos x\; dx

Put sin\; x = t \Rightarrow cos\; x\; dx = dt

When x = 0, t = 0 and\; when \; x =\frac{\pi }{2}, t =1

\begin{aligned} \mathrm{I} &=\int_{0}^{1} e^{t} \\\\ &=\left[e^{t}\right]_{0}^{1} \\\\ &=e^{1}-e^{0} \\\\ &=\mathrm{e}-1 \end{aligned}


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