ExpandExpandProcessLogApply the sum rule of integration: \n \( \mathrm { \displaystyle\int ( f ( u ) + g ( u ) ) du = \displaystyle\int f ( u ) du + \displaystyle\int g ( u ) du} \)ExpandStr1. STEP:Str\( \mathrm { \displaystyle\int\limits_{\displaystyle\frac{\pi}{4}}^{\displaystyle\frac{5 \cdot \pi}{4}} \sin{\left(x \right)}\, dx } \)ProcessLogUse the common integral rule: \n \( \mathrm { \displaystyle\int \sin u du = - \cos u + c } \)Str\( \mathrm { \displaystyle\int\limits_{\displaystyle\frac{\pi}{4}}^{\displaystyle\frac{5 \cdot \pi}{4}} \sin{\left(x \right)}\, dx = - \cos{\left(x \right)} \Bigg|_{ \displaystyle\frac{\pi}{4} } ^ { \displaystyle\frac{5 \cdot \pi}{4} } } \)ProcessLogOrganize the integrals according to upper and lower limits. \n Upper limit: \n \( \mathrm { x = \displaystyle\frac{5 \cdot \pi}{4} \Longrightarrow - \cos{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} } \) \n Lower limit: \n \( \mathrm { x = \displaystyle\frac{\pi}{4} \Longrightarrow - \cos{\left(\displaystyle\frac{\pi}{4} \right)} } \) \n The value of the upper limit is subtracted from the value of the lower limit.Str\( \mathrm { - \cos{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} + \cos{\left(\displaystyle\frac{\pi}{4} \right)} } \)StrProcessLogSubstitute backStr\( \mathrm { - \cos{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} + \cos{\left(\displaystyle\frac{\pi}{4} \right)} + \displaystyle\int\limits_{\displaystyle\frac{\pi}{4}}^{\displaystyle\frac{5 \cdot \pi}{4}} \left(- \cos{\left(x \right)}\right)\, dx } \)Str2. STEP:Str\( \mathrm { \displaystyle\int\limits_{\displaystyle\frac{\pi}{4}}^{\displaystyle\frac{5 \cdot \pi}{4}} \left(- \cos{\left(x \right)}\right)\, dx } \)ProcessLogApply the integral constant multiple rule: \n \( \mathrm { \displaystyle\int a \cdot f ( u ) du = a \displaystyle\int f ( u ) du } \)Str\( \mathrm { - \displaystyle\int\limits_{\displaystyle\frac{\pi}{4}}^{\displaystyle\frac{5 \cdot \pi}{4}} \cos{\left(x \right)}\, dx } \)ProcessLogUse the common integral rule: \n \( \mathrm { \displaystyle\int \cos u du = \sin u + c } \)Str\( \mathrm { - \displaystyle\int\limits_{\displaystyle\frac{\pi}{4}}^{\displaystyle\frac{5 \cdot \pi}{4}} \cos{\left(x \right)}\, dx = - \sin{\left(x \right)} \Bigg|_{ \displaystyle\frac{\pi}{4} } ^ { \displaystyle\frac{5 \cdot \pi}{4} } } \)ProcessLogOrganize the integrals according to upper and lower limits. \n Upper limit: \n \( \mathrm { x = \displaystyle\frac{5 \cdot \pi}{4} \Longrightarrow - \sin{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} } \) \n Lower limit: \n \( \mathrm { x = \displaystyle\frac{\pi}{4} \Longrightarrow - \sin{\left(\displaystyle\frac{\pi}{4} \right)} } \) \n The value of the upper limit is subtracted from the value of the lower limit.Str\( \mathrm { - \sin{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} + \sin{\left(\displaystyle\frac{\pi}{4} \right)} } \)StrProcessLogSubstitute backStr\( \mathrm { - \cos{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} + \cos{\left(\displaystyle\frac{\pi}{4} \right)} - \sin{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} + \sin{\left(\displaystyle\frac{\pi}{4} \right)} } \)\( \mathrm { \displaystyle\int\limits_{\displaystyle\frac{\pi}{4}}^{\displaystyle\frac{5 \cdot \pi}{4}} \sin{\left(x \right)}\, dx + \displaystyle\int\limits_{\displaystyle\frac{\pi}{4}}^{\displaystyle\frac{5 \cdot \pi}{4}} \left(- \cos{\left(x \right)}\right)\, dx } \)\( \mathrm { - \cos{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} + \cos{\left(\displaystyle\frac{\pi}{4} \right)} - \sin{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} + \sin{\left(\displaystyle\frac{\pi}{4} \right)} } \)Evaluate the integrals\( \mathrm { \displaystyle\int\limits_{\displaystyle\frac{\pi}{4}}^{\displaystyle\frac{5 \cdot \pi}{4}} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)\, dx } \)\( \mathrm { - \cos{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} + \cos{\left(\displaystyle\frac{\pi}{4} \right)} - \sin{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} + \sin{\left(\displaystyle\frac{\pi}{4} \right)} } \)Evaluate the integralProcessLogExpress the value of the trigonometric function: \n \( \mathrm { \cos{\left(\displaystyle\frac{\pi}{4} \right)} = \displaystyle\frac{\sqrt{2}}{2} } \)Str\( \mathrm { - \cos{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} + \displaystyle\frac{\sqrt{2}}{2} - \sin{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} + \sin{\left(\displaystyle\frac{\pi}{4} \right)} } \)ProcessLogExpress the value of the trigonometric function: \n \( \mathrm { \sin{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} = - \displaystyle\frac{\sqrt{2}}{2} } \)Str\( \mathrm { - \cos{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} + \displaystyle\frac{\sqrt{2}}{2} + \displaystyle\frac{\sqrt{2}}{2} + \sin{\left(\displaystyle\frac{\pi}{4} \right)} } \)ProcessLogExpress the value of the trigonometric function: \n \( \mathrm { \cos{\left(\displaystyle\frac{5 \cdot \pi}{4} \right)} = - \displaystyle\frac{\sqrt{2}}{2} } \)Str\( \mathrm { \displaystyle\frac{\sqrt{2}}{2} + \displaystyle\frac{\sqrt{2}}{2} + \displaystyle\frac{\sqrt{2}}{2} + \sin{\left(\displaystyle\frac{\pi}{4} \right)} } \)ProcessLogExpress the value of the trigonometric function: \n \( \mathrm { \sin{\left(\displaystyle\frac{\pi}{4} \right)} = \displaystyle\frac{\sqrt{2}}{2} } \)Str\( \mathrm { \displaystyle\frac{\sqrt{2}}{2} + \displaystyle\frac{\sqrt{2}}{2} + \displaystyle\frac{\sqrt{2}}{2} + \displaystyle\frac{\sqrt{2}}{2} } \)ProcessLog\( \mathrm { \displaystyle\frac{\sqrt{2}}{2} + \displaystyle\frac{\sqrt{2}}{2} + \displaystyle\frac{\sqrt{2}}{2} + \displaystyle\frac{\sqrt{2}}{2} = 2 \cdot \sqrt{2} } \)Str\( \mathrm { 2 \cdot \sqrt{2} } \)\( \mathrm { \displaystyle\int\limits_{\displaystyle\frac{\pi}{4}}^{\displaystyle\frac{5 \cdot \pi}{4}} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)\, dx } \)\( \mathrm { 2 \cdot \sqrt{2} } \)SimplifyProcessLogDecimal form: \[ 2.82842712474619 \]

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Solution: Find the area enclosed by the curves y = sin(x) and y = cos(x) on the interval [π/4, 5π/4]

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