Question 2 - Case Based Questions (MCQ) - Chapter 7 Class 12 Integrals

Transcript

Question 2 ∫1▒〖𝑒^𝑥 [𝑓(𝑥)+𝑓^′ (𝑥)] 〗 𝑑𝑥 = ∫1▒〖𝒆^𝒙 𝒇(𝒙)𝒅𝒙〗+∫1▒〖𝑒^𝑥 𝑓′(𝑥)𝑑𝑥〗 Using integration by parts = 𝒇(𝒙) ∫1▒𝒆^𝒙 𝒅𝒙−∫1▒〖𝒇^′ (𝒙) 𝒆^𝒙 〗 𝒅𝒙 + ∫1▒〖𝑓^′ (𝑥) 𝑒^𝑥 〗 𝑑𝑥 = 𝑓(𝑥) 𝑒^𝑥−∫1▒〖𝑓^′ (𝑥) 𝑒^𝑥 〗 𝑑𝑥 + ∫1▒〖𝑓^′ (𝑥) 𝑒^𝑥 〗 𝑑𝑥 = 𝑒^𝑥 𝑓(𝑥)+𝐶 Based on the above information, answer any four of the following questions. Question 1 ∫1▒〖𝑒^𝑥 (sin⁡〖𝑥 〗+cos⁡𝑥)〗 𝑑𝑥=______________. (a) 𝑒^𝑥 cos⁡𝑥+𝑐 (b) 𝑒^𝑥 sin⁡𝑥+𝑐 (c) 𝑒^𝑥+𝑐 (d) 𝑒^𝑥 (−cos⁡𝑥+sin⁡𝑥 )+c ∫1▒〖𝑒^𝑥 (sin⁡〖𝑥 〗+cos⁡𝑥)〗 𝑑𝑥 Putting 𝒇(𝒙)=𝒔𝒊𝒏 𝒙" " ∴ 𝒇^′ (𝒙)=cos⁡𝑥 Thus, ∫1▒〖𝑒^𝑥 (sin⁡〖𝑥 〗+cos⁡𝑥)〗 𝑑𝑥=𝒆^𝒙 𝒔𝒊𝒏 𝒙+𝑪 So, the correct answer is (b) Question 2 ∫1▒〖𝑒^𝑥 ((𝑥 − 1)/𝑥^2 ) 〗 𝑑𝑥=______________. (a) 𝑒^𝑥+𝑐 (b) 𝑒^𝑥/𝑥+𝑐 (c) 𝑒^𝑥/𝑥^2 +𝑐 (d) 〖−𝑒〗^𝑥/𝑥^2 +𝑐 ∫1▒〖𝑒^𝑥 ((𝑥 − 1)/𝑥^2 ) 〗 𝑑𝑥= ∫1▒〖𝑒𝑥" " (𝑥/𝑥^2 − 1/𝑥2) 𝑑𝑥〗 = ∫1▒〖𝒆𝒙" " (𝟏/𝒙 − 𝟏/𝒙𝟐) 𝒅𝒙〗 Putting 𝒇(𝒙)=𝟏/𝒙 ∴ 𝒇^′ (𝒙)=(−1)/𝑥^2 Thus, ∫1▒〖𝑒𝑥" " (1/𝑥 − 1/𝑥2) 𝑑𝑥〗 = 𝒆^𝒙/𝒙+𝑪 So, the correct answer is (b) Question 3 ∫1▒〖𝑒^𝑥 (1+𝑥) 〗 𝑑𝑥=______________. (a) 𝑥𝑒^𝑥+𝑐 (b) 𝑒^𝑥+𝑐 (c) 𝑒^(−𝑥)+𝑐 (d) none of these ∫1▒〖𝑒^𝑥 (1+𝑥)〗 𝑑𝑥=∫1▒〖𝒆^𝒙 (𝒙+𝟏)〗 𝒅𝒙 Putting 𝒇(𝒙)=𝒙 ∴ 𝒇^′ (𝒙)=1 Thus, ∫1▒〖𝑒^𝑥 (𝑥+1)〗 𝑑𝑥=𝒆^𝒙 × 𝒙+𝑪 So, the correct answer is (a) Question 4 ∫1_0^𝜋▒𝑒^𝑥 (tan⁡𝑥+sec^2⁡𝑥) 𝑑𝑥 = _________. (a) 0 (b) 1 (c) −1 (d) −𝑒^𝜋 ∫1_0^𝜋▒𝑒^𝑥 (tan⁡𝑥+sec^2⁡𝑥) 𝑑𝑥 Putting 𝒇(𝒙)=𝒕𝒂𝒏 𝒙" " ∴ 𝒇^′ (𝒙)=〖𝑠𝑒𝑐〗^2⁡𝑥 Thus, ∫1_0^𝜋▒𝑒^𝑥 (tan⁡𝑥+sec^2⁡𝑥) 𝑑𝑥 = 〖[𝒆^𝒙 𝒕𝒂𝒏 𝒙]〗_0^𝜋 =[𝒆^𝒙 𝒕𝒂𝒏 𝜋−𝒆^𝒙 𝒕𝒂𝒏 𝟎] =[𝒆^𝒙 𝒕𝒂𝒏 𝜋−𝒆^𝒙 𝒕𝒂𝒏 𝟎] =[𝒆^𝒙 × 𝟎−𝒆^𝒙 × 𝟎] =0−0 =0 So, the correct answer is (a) Question 5 ∫1▒〖〖𝑥𝑒〗^𝑥/(1 + 𝑥)^2 〗 𝑑𝑥=______________. (a) 𝑥𝑒^𝑥+𝑐 (b) 𝑒^𝑥/(𝑥 + 1)^2 +𝑐 (c) (𝑥 𝑒^𝑥)/(𝑥 + 1)+𝑐 (d) 𝑒^𝑥/(𝑥 + 1)+𝑐 ∫1▒〖𝑒^𝑥 ((𝑥 )/(1 + 𝑥)^2 ) 〗 𝑑𝑥= ∫1▒〖𝑒𝑥" " ((1 + 𝑥 − 1 )/(1 + 𝑥)^2 ) 𝑑𝑥〗 =" " ∫1▒〖𝑒𝑥" " ((1 + 𝑥 )/(1 + 𝑥)^2 −1/(1 + 𝑥)^2 ) 𝑑𝑥〗 =" " ∫1▒〖𝒆𝒙" " (𝟏/((𝟏 + 𝒙))−𝟏/(𝟏 + 𝒙)^𝟐 ) 𝒅𝒙〗 It is of form ∫1▒〖𝒆^𝒙 (𝒇(𝒙)+𝒇^′ (𝒙)) 〗 𝒅𝒙=𝑒^𝑥 𝑓(𝑥)+𝐶 Putting 𝒇(𝒙)=𝟏/((𝟏 + 𝒙)) ∴ 𝒇^′ (𝒙)=(−1)/(𝟏 + 𝒙)^2 Thus, ∫1▒〖𝑒𝑥" " (1/((1 + 𝑥))−1/(1 + 𝑥)^2 ) 𝑑𝑥〗 = 𝒆^𝒙/((𝟏 + 𝒙))+𝑪 So, the correct answer is (d)