Example 21 - Chapter 7 Class 12 Integrals

Transcript

Example 21 Find ∫1▒𝑒^𝑥 sin⁡𝑥 𝑑𝑥 Let I1 = ∫1▒〖 𝑒^𝑥 〗 sin⁡𝑥 𝑑𝑥 I1 = sin⁡𝑥 ∫1▒〖𝑒^𝑥 𝑑𝑥〗−∫1▒(𝑑(sin⁡𝑥 )/𝑑𝑥 ∫1▒〖𝑒^𝑥 𝑑𝑥〗) 𝑑𝑥 I1 = 𝑒^𝑥 sin⁡𝑥−∫1▒〖cos⁡𝑥 . 𝑒^𝑥 𝑑𝑥〗 Now we know that ∫1▒〖𝑓(𝑥) 𝑔⁡(𝑥) 〗 𝑑𝑥=𝑓(𝑥) ∫1▒𝑔(𝑥) 𝑑𝑥−∫1▒(𝑓′(𝑥)∫1▒𝑔(𝑥) 𝑑𝑥) 𝑑𝑥 Putting f(x) = sin x and g(x) = ex Solving I2 I2 = ∫1▒〖cos⁡𝑥 . 𝑒^𝑥 𝑑𝑥〗 I2 = cos x ∫1▒〖𝑒^𝑥 𝑑𝑥〗 – ∫1▒〖((cos⁡𝑥)′〗 ∫1▒〖𝑒^𝑥 𝑑𝑥〗)𝑑𝑥 I2 = cos x 𝑒^𝑥 – ∫1▒〖(−sin⁡𝑥)〗 𝑒^𝑥 𝑑𝑥 I2 = 𝑒^𝑥 cos x + ∫1▒sin⁡𝑥 𝑒^𝑥 𝑑𝑥 I2 = 𝑒^𝑥 cos x + 𝐼1 Now we know that ∫1▒〖𝑓(𝑥) 𝑔⁡(𝑥) 〗 𝑑𝑥=𝑓(𝑥) ∫1▒𝑔(𝑥) 𝑑𝑥−∫1▒(𝑓′(𝑥)∫1▒𝑔(𝑥) 𝑑𝑥) 𝑑𝑥 Putting f(x) = sin x and g(x) = ex Now, Putting value of I2 in (1) , I1 = " " 𝑒^𝑥 sin⁡𝑥−∫1▒〖cos⁡𝑥 𝑒^𝑥 〗 𝑑𝑥 I1 = " " 𝑒^𝑥 sin⁡𝑥−(𝑒^𝑥 cos⁡𝑥+𝐼1)+𝐶 I1 = " " 𝑒^𝑥 sin⁡𝑥−𝑒^𝑥 cos⁡𝑥−𝐼1+𝐶 2I1 = " " 𝑒^𝑥 sin⁡𝑥−𝑒^𝑥 cos⁡𝑥 + 𝐶 I1 = 1/2 (𝑒^𝑥 sin⁡𝑥−𝑒^𝑥 cos⁡𝑥 ) + C 𝑰𝟏 = 𝒆^𝒙/𝟐 (𝒔𝒊𝒏⁡𝒙−𝒄𝒐𝒔⁡𝒙 ) + C