# Misc 27 - Chapter 7 Class 12 Integrals

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Misc 27 Evaluate the definite integral 0𝜋2cos2𝑥 𝑑𝑥cos2𝑥 + 4sin2𝑥 Let I = 0𝜋2𝑐𝑜𝑠2𝑥𝑐𝑜𝑠2𝑥 + 4𝑠𝑖𝑛2𝑥 𝑑𝑥 = 0𝜋2𝑐𝑜𝑠2𝑥𝑐𝑜𝑠2𝑥 + 4 1 − 𝑐𝑜𝑠2𝑥 𝑑𝑥 = 0𝜋2𝑐𝑜𝑠2𝑥4 − 3 𝑐𝑜𝑠2𝑥 𝑑𝑥 = −13 0𝜋4 −3 𝑐𝑜𝑠2 𝑥 4 − 3 𝑐𝑜𝑠2𝑥 𝑑𝑥 = −130𝜋24 −3 𝑐𝑜𝑠2𝑥 −44 − 3 𝑐𝑜𝑠2𝑥 𝑑𝑥 = −130𝜋21−44 − 3 𝑐𝑜𝑠2𝑥 𝑑𝑥 = −130𝜋2𝑑𝑥+430𝜋2𝑑𝑥4 − 3 𝑐𝑜𝑠2𝑥 Dividing numerator and denominator by 𝑐𝑜𝑠2 𝑥 = −13𝜋2+43 0𝜋2𝑑𝑥𝑐𝑜𝑠2 𝑥4 − 3 𝑐𝑜𝑠2𝑥 𝑐𝑜𝑠2 𝑥 𝑑𝑥 = −13𝜋2+43 0𝜋2𝑠𝑒𝑐2𝑥4 𝑠𝑒𝑐2𝑥 − 3 𝑑𝑥 = −𝜋6+43 0𝜋2𝑠𝑒𝑐2𝑥4 (1 + 𝑡𝑎𝑛2𝑥) − 3 𝑑𝑥 Put tan x = t so that 𝑠𝑒𝑐2 x dx = dt Thus, When x = 0, t = 0, and when x = 𝜋2, 𝑡= Substituting values and limit ∴ I =−𝜋6+43 0uc1𝑑𝑡4 1+ 𝑡2 −3 =−𝜋6+43 0uc1𝑑𝑡4𝑡2+1 =−𝜋6+43 ∙14 0uc1𝑑𝑡 𝑡2+14 =𝜋6+43∙14 × 112 𝑡𝑎𝑛−1𝑡120uc1 = −𝜋6+23∙ 𝑡𝑎𝑛−1 2𝑡0uc1 =−𝜋6+23∙[𝑡𝑎𝑛−1uc1−𝑡𝑎𝑛−10] = −𝜋6+23∙𝜋2−0 =−𝜋6+𝜋3 =𝝅𝟔

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Misc 27 You are here

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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.