# Misc 25 - Chapter 7 Class 12 Integrals

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Misc 25 Evaluate the definite integral 𝜋2𝜋e𝑥1 −sin𝑥1 −cos𝑥 𝑑𝑥 𝜋2𝜋e𝑥1 − sin𝑥1 − cos𝑥 𝑑𝑥 = 𝜋2𝜋e𝑥1 1 − cos𝑥−sin𝑥1 − cos𝑥 𝑑𝑥 Let f(x) = −sin𝑥1 − cos𝑥 f’(x) = −𝑐𝑜𝑠𝑥 (1 − cos𝑥) − (sin𝑥) (sin𝑥)(1 − cos𝑥)2 =−cos𝑥 − 𝑐𝑜𝑠2𝑥 − 𝑠𝑖𝑛2𝑥(1 − cos 𝑥)2 = −cos𝑥 − 𝑐𝑜𝑠2𝑥 + 𝑠𝑖𝑛2𝑥(1 − 𝑐𝑜𝑠 𝑥)2 = cos𝑥 − 1(1 −cos𝑥)2 = − 1 cos𝑥 (1 −cos𝑥)2 = 11 − cos𝑥 Hence the given integration is of form, 𝑒𝑥(𝑓𝑥+𝑓′𝑥) 𝑑𝑥= 𝑒𝑥𝑓(𝑥) Where f(x) = −sin𝑥1 −cos𝑥 f’(x) = 11 − cos𝑥 Hence, 𝜋2𝜋e𝑥1 − sin𝑥1 − cos𝑥 𝑑𝑥 = 𝜋2𝜋e𝑥1 1 − cos𝑥−sin𝑥1 − cos𝑥 𝑑𝑥 = 𝑒𝑥 𝑓(𝑥)𝜋2𝜋 = 𝑒𝑥 −sin𝑥1 −cos𝑥𝜋2𝜋 = 𝑒𝑥sin𝑥cos𝑥 −1𝜋2𝜋 Putting limits, = 𝑒𝜋sin𝜋cos𝜋 −1− 𝑒𝜋2 sin𝜋2cos 𝜋2 −1 = 𝑒𝜋(0)−1 −1− 𝑒𝜋2 (1)0 − 1 = 𝒆𝝅𝟐

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Integration Formula Sheet - Chapter 7 Class 12 Formulas Important

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.