SATHEE: Indefinite Integration 1 Question 69

Indefinite Integration 1 Question 69

70. Evaluate $\int_{0}^{π} \frac{e^{\cos x}}{e^{\cos x} + e^{- \cos x}} d x$ .

$(1999, 3 M)$

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Solution:

Let $I = \int_{0}^{π} \frac{e^{\cos x}}{e^{\cos x} + e^{- \cos x}} d x$

$\begin{aligned} = \int_{0}^{π} \frac{e^{\cos (π - x)}}{e^{\cos (π - x)} + e^{- \cos (π - x)}} d x \\ [∵ \int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x] \\ \Rightarrow I & = \int_{0}^{π} \frac{e^{- \cos x}}{e^{- \cos x} + e^{\cos x}} d x \end{aligned}$

On adding Eqs. (i) and (ii), we get

$\begin{aligned} = \int_{0}^{π} \frac{e^{\cos x} + e^{- \cos x}}{e^{\cos x} + e^{- \cos x}} d x = \int_{0}^{π} 1 d x = [x]_{0}^{π} = π \\ \Rightarrow I & = π / 2 \end{aligned}$

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Indefinite Integration 1 Question 69

70. Evaluate ∫0πecos⁡xecos⁡x+e−cos⁡xdx.

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70. Evaluate $\int_{0}^{π} \frac{e^{\cos x}}{e^{\cos x} + e^{- \cos x}} d x$ .