# Misc 31 - Chapter 7 Class 12 Integrals

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Misc 31 Evaluate the definite integral 0𝜋2sin2𝑥tan−1sin𝑥 𝑑𝑥 0𝜋2sin2𝑥tan−1sin𝑥 𝑑𝑥 = 0𝜋22sin𝑥cos𝑥 tan−1sin𝑥 𝑑𝑥 Let sin𝑥=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 cos𝑥=𝑑𝑡𝑑𝑥 𝑑𝑥=𝑑𝑡cos𝑥 Substituting x and dx 0𝜋22sin𝑥cos𝑥 𝑡𝑎𝑛−1 (sin𝑥) 𝑑𝑥 = 012𝑡cos𝑥 𝑡𝑎𝑛−1 (𝑡) 𝑑𝑡𝑐𝑜𝑠𝑥 = 012𝑡 𝑡𝑎𝑛−1𝑡 𝑑𝑡 = 201𝑡 𝑡𝑎𝑛−1 𝑡 𝑑𝑡 =2𝑡𝑎𝑛−1𝑡𝑡 𝑑𝑡 − 𝑑 𝑡𝑎𝑛−1𝑑𝑡 𝑡 𝑑𝑡 𝑑𝑡 = 2𝑡𝑎𝑛−1 𝑡 𝑡22−11 + 𝑡2×𝑡22 𝑑𝑡 = 2𝑡22𝑡𝑎𝑛−1 𝑡−12𝑡22 𝑑𝑡 = 𝑡2 𝑡𝑎𝑛−1 𝑡−𝑡21 + 𝑡2 𝑑𝑡 Solving 𝑰𝟏 I1 = 𝑡21 + 𝑡2𝑑𝑡 Adding and Subtracting 1 in numerator. I1 = 𝑡2 + 1 − 1𝑡2 + 1𝑑𝑡 I1= 𝑡2 + 1𝑡2 + 1−1𝑡2 + 1𝑑𝑡 I1= 1−1𝑡2 + 1 I1= 𝑑𝑡−𝑑𝑡𝑡2 + 1 I1= t − 𝑡𝑎𝑛−1 (t) Thus, our equation becomes ∴ 𝑡𝑎𝑛−1 𝑡×𝑡 𝑑𝑡= 𝑡2 𝑡𝑎𝑛−1−( I1) = 𝑡2𝑡𝑎𝑛−1 𝑡−𝑡−𝑡𝑎𝑛−1𝑡 = 𝑡2𝑡𝑎𝑛−1 𝑡−𝑡+𝑡𝑎𝑛−1𝑡 =𝐹(𝑥) 201𝑡𝑎𝑛−1 𝑡 𝑡 𝑑𝑡 =𝐹1−𝐹(0) =1×𝑡𝑎𝑛−1 1−1+𝑡𝑎𝑛−11−0+𝑡𝑎𝑛−10 = 𝜋4−1+𝜋4−0− 0+0 = 𝜋2−1−0 = 𝝅𝟐−𝟏

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Misc 8 Important

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Misc 18 Important

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Misc 30 Important

Misc 31 You are here

Misc 32 Important

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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.