Ex 7.9, 20 - Direct Integrate (x ex + sin pi x / 4) dx - Definate Integration - By Formulae

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Ex7.9, 20 ∫_0^1β–’(π‘₯ 𝑒^π‘₯+sinβ‘γ€–πœ‹π‘₯/4γ€— ) 𝑑π‘₯ Step 1 :- Let F(π‘₯)=∫1β–’(π‘₯𝑒^π‘₯+𝑠𝑖𝑛 πœ‹/4 π‘₯)𝑑π‘₯ =∫1β–’γ€–π‘₯𝑒^π‘₯ 𝑑π‘₯+∫1▒〖𝑠𝑖𝑛 πœ‹/4 π‘₯ 𝑑π‘₯γ€—γ€— Solving π‘°πŸ ∫1β–’γ€–π‘₯𝑒^π‘₯ 𝑑π‘₯γ€— Integration by parts ∫1β–’γ€–π‘₯𝑒^π‘₯ 𝑑π‘₯=π‘₯∫1▒〖𝑒^π‘₯ 𝑑π‘₯βˆ’βˆ«1β–’[(𝑑π‘₯/𝑑π‘₯) ∫1▒〖𝑒^π‘₯ 𝑑π‘₯γ€—]𝑑π‘₯γ€—γ€— =π‘₯𝑒^π‘₯βˆ’βˆ«1β–’(+1.𝑒^π‘₯ 𝑑π‘₯)𝑑π‘₯ =π‘₯𝑒^π‘₯βˆ’βˆ«1▒〖𝑒^π‘₯ 𝑑π‘₯γ€— =π‘₯𝑒^π‘₯βˆ’π‘’^π‘₯ =𝑒^π‘₯ (π‘₯βˆ’1) Therefore, F(π‘₯)=∫1β–’γ€–π‘₯𝑒^π‘₯ 𝑑π‘₯+∫1▒〖𝑠𝑖𝑛 πœ‹/4 π‘₯ 𝑑π‘₯γ€—γ€— =𝑒^π‘₯ (π‘₯βˆ’1)+1/(πœ‹/4) (βˆ’π‘π‘œπ‘  πœ‹π‘₯/4) =𝑒^π‘₯ (π‘₯βˆ’1)βˆ’4/πœ‹ π‘π‘œπ‘  πœ‹π‘₯/4 Hence, F(π‘₯)=𝑒^π‘₯ (π‘₯βˆ’1)βˆ’4/πœ‹ π‘π‘œπ‘  πœ‹π‘₯/4 Step 2 :- ∫_0^1β–’(π‘₯𝑒^π‘₯+𝑠𝑖𝑛 πœ‹π‘₯/4) 𝑑π‘₯=𝐹(1)βˆ’πΉ(0) =𝑒^1 (1βˆ’1)βˆ’4/πœ‹ π‘π‘œπ‘  ((πœ‹Γ—1))/4βˆ’π‘’^0 (0βˆ’1)+4/πœ‹ π‘π‘œπ‘  ((πœ‹Γ—0))/4 =𝑒×0βˆ’4/πœ‹ π‘π‘œπ‘  ((πœ‹))/4βˆ’1(βˆ’1)+4/πœ‹ cos⁑0 =(βˆ’4)/( πœ‹) π‘π‘œπ‘  ((πœ‹))/4+1+4/πœ‹ =(βˆ’4)/( πœ‹) 1/√2+1+4/πœ‹ =𝟏+πŸ’/π…βˆ’(𝟐√𝟐)/𝝅

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