int_0^{pi /4} (cos x - sin x) dx + int_{pi /4 | vedclass

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(D) Let $I = \int_0^{\pi /4} (\cos x - \sin x) dx + \int_{\pi /4}^{5\pi /4} (\sin x - \cos x) dx + \int_{2\pi }^{\pi /4} (\cos x - \sin x) dx$. Evalua

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$4\sqrt{2} - 2$

Vedclass · Answer

(D) Let $I = \int_0^{\pi /4} (\cos x - \sin x) dx + \int_{\pi /4}^{5\pi /4} (\sin x - \cos x) dx + \int_{2\pi }^{\pi /4} (\cos x - \sin x) dx$. Evaluating each integral: $1$. $\int_0^{\pi /4} (\cos x - \sin x) dx = [\sin x + \cos x]_0^{\pi /4} = (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}) - (0 + 1) = \sqrt{2} - 1$. $2$. $\int_{\pi /4}^{5\pi /4} (\sin x - \cos x) dx = [-\cos x - \sin x]_{\pi /4}^{5\pi /4} = -[\cos x + \sin x]_{\pi /4}^{5\pi /4} = -[(-\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}) - (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}})] = -[-\sqrt{2} - \sqrt{2}] = 2\sqrt{2}$. $3$. $\int_{2\pi }^{\pi /4} (\cos x - \sin x) dx = [\sin x + \cos x]_{2\pi }^{\pi /4} = (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}) - (0 + 1) = \sqrt{2} - 1$. Summing these values: $I = (\sqrt{2} - 1) + 2\sqrt{2} + (\sqrt{2} - 1) = 4\sqrt{2} - 2$.

$\int_0^{\pi /4} (\cos x - \sin x) dx + \int_{\pi /4}^{5\pi /4} (\sin x - \cos x) dx + \int_{2\pi }^{\pi /4} (\cos x - \sin x) dx$ is equal to

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