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Example 21 - Find integral ex sin x dx - Class 12 NCERT - Examples

Example 21 - Chapter 7 Class 12 Integrals - Part 2
Example 21 - Chapter 7 Class 12 Integrals - Part 3

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Example 21 Find ∫1▒𝑒^π‘₯ sin⁑π‘₯ 𝑑π‘₯ Let I1 = ∫1β–’γ€– 𝑒^π‘₯ γ€— sin⁑π‘₯ 𝑑π‘₯ I1 = sin⁑π‘₯ ∫1▒〖𝑒^π‘₯ 𝑑π‘₯γ€—βˆ’βˆ«1β–’(𝑑(sin⁑π‘₯ )/𝑑π‘₯ ∫1▒〖𝑒^π‘₯ 𝑑π‘₯γ€—) 𝑑π‘₯ I1 = 𝑒^π‘₯ sin⁑π‘₯βˆ’βˆ«1β–’γ€–cos⁑π‘₯ . 𝑒^π‘₯ 𝑑π‘₯γ€— Now we know that ∫1▒〖𝑓(π‘₯) 𝑔⁑(π‘₯) γ€— 𝑑π‘₯=𝑓(π‘₯) ∫1▒𝑔(π‘₯) 𝑑π‘₯βˆ’βˆ«1β–’(𝑓′(π‘₯)∫1▒𝑔(π‘₯) 𝑑π‘₯) 𝑑π‘₯ Putting f(x) = sin x and g(x) = ex Solving I2 I2 = ∫1β–’γ€–cos⁑π‘₯ . 𝑒^π‘₯ 𝑑π‘₯γ€— I2 = cos x ∫1▒〖𝑒^π‘₯ 𝑑π‘₯γ€— – ∫1β–’γ€–((cos⁑π‘₯)β€²γ€— ∫1▒〖𝑒^π‘₯ 𝑑π‘₯γ€—)𝑑π‘₯ I2 = cos x 𝑒^π‘₯ – ∫1β–’γ€–(βˆ’sin⁑π‘₯)γ€— 𝑒^π‘₯ 𝑑π‘₯ I2 = 𝑒^π‘₯ cos x + ∫1β–’sin⁑π‘₯ 𝑒^π‘₯ 𝑑π‘₯ I2 = 𝑒^π‘₯ cos x + 𝐼1 Now we know that ∫1▒〖𝑓(π‘₯) 𝑔⁑(π‘₯) γ€— 𝑑π‘₯=𝑓(π‘₯) ∫1▒𝑔(π‘₯) 𝑑π‘₯βˆ’βˆ«1β–’(𝑓′(π‘₯)∫1▒𝑔(π‘₯) 𝑑π‘₯) 𝑑π‘₯ Putting f(x) = sin x and g(x) = ex Now, Putting value of I2 in (1) , I1 = " " 𝑒^π‘₯ sin⁑π‘₯βˆ’βˆ«1β–’γ€–cos⁑π‘₯ 𝑒^π‘₯ γ€— 𝑑π‘₯ I1 = " " 𝑒^π‘₯ sin⁑π‘₯βˆ’(𝑒^π‘₯ cos⁑π‘₯+𝐼1)+𝐢 I1 = " " 𝑒^π‘₯ sin⁑π‘₯βˆ’π‘’^π‘₯ cos⁑π‘₯βˆ’πΌ1+𝐢 2I1 = " " 𝑒^π‘₯ sin⁑π‘₯βˆ’π‘’^π‘₯ cos⁑π‘₯ + 𝐢 I1 = 1/2 (𝑒^π‘₯ sin⁑π‘₯βˆ’π‘’^π‘₯ cos⁑π‘₯ ) + C π‘°πŸ = 𝒆^𝒙/𝟐 (π’”π’Šπ’β‘π’™βˆ’π’„π’π’”β‘π’™ ) + C

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.