Misc 26 - Chapter 7 Class 12 Integrals
Last updated at Dec. 23, 2019 by Teachoo
Transcript
Misc 26 Evaluate the definite integral β«_0^(π/4)βγ(sinβ‘π₯ cosβ‘π₯)/(cos^4β‘π₯ + sin^4β‘π₯ ) ππ₯γ β«_0^(π/4)βγ(sinβ‘π₯ cosβ‘π₯)/(cos^4β‘π₯ + sin^4β‘π₯ ) ππ₯γ Adding and subtracting 2 γπ ππγ^2 π₯ γπππ γ^2 π₯ from denominator = β«_0^(π/4)βγ(sinβ‘π₯ cosβ‘π₯)/(cos^4β‘π₯ + sin^4β‘γπ₯ + 2γπ ππγ^2 π₯ γπππ γ^2 π₯ β 2γπ ππγ^2 π₯ γπππ γ^2 π₯γ ) ππ₯γ = β«_0^(π/4)βγ(sinβ‘π₯ cosβ‘π₯)/(γγ(cosγ^2β‘π₯ + sin^2β‘π₯)γ^2 β 2γπ ππγ^2 π₯ γπππ γ^2 π₯) ππ₯γ = β«_0^(π/4)βγ(sinβ‘π₯ cosβ‘π₯)/(1 β (4 γπ ππγ^2 π₯ γπππ γ^2 π₯)/2) ππ₯γ = β«_0^(π/4)βγ(sinβ‘π₯ cosβ‘π₯)/(1 β 1/2 (2 sinβ‘π₯ cosβ‘π₯ )^2 ) ππ₯γ = β«_0^(π/4)βγ(sinβ‘π₯ cosβ‘π₯)/(1 β (γπ ππγ^2 2π₯)/2) ππ₯γ = β«_0^(π/4)βγ(2 sinβ‘π₯ cosβ‘π₯)/(2 β γπ ππγ^(2 ) 2π₯) ππ₯γ = β«_0^(π/4)βγsinβ‘2π₯/(1 + (1 β γπ ππγ^(2 ) 2π₯)) ππ₯γ = β«_0^(π/4)βγsinβ‘2π₯/(1 + γπππ γ^2 2π₯) γ ππ₯ (γπππ γ^2 π₯+γπ ππγ^2 π₯=1" " ) ("2 sin " π₯ cosβ‘π₯=sinβ‘2π₯ " " ) ("2 sin " π₯ cosβ‘π₯=sinβ‘2π₯ " " ) Let t = cos 2π₯ ππ‘/ππ₯=β2 sinβ‘2π₯ (βππ‘)/2=sinβ‘2π₯ ππ₯ So, our equation becomes β«_0^(π/4)βγsinβ‘2π₯/(1 + γπππ γ^2 2π₯) γ ππ₯ = (β1)/2 β«1_1^0βππ‘/(1 + π‘^2 ) = (β1)/2 [γπ‘ππγ^(β1) (π‘)]_1^0 = (β1)/2 [γπ‘ππγ^(β1) (0)βγπ‘ππγ^(β1) (1)] = (β1)/2 (0βπ/4) = π /π
Miscellaneous
Misc 1
Important
Misc 2 Important
Misc 3 Important
Misc 4
Misc 5 Important
Misc 6
Misc 7 Important
Misc 8 Important
Misc 9
Misc 10 Important
Misc 11
Misc 12
Misc 13
Misc 14 Important
Misc 15
Misc 16
Misc 17
Misc 18 Important
Misc 19 Important
Misc 20 Important
Misc 21
Misc 22
Misc 23
Misc 24 Important
Misc 25 Important
Misc 26 Important You are here
Misc 27 Important
Misc 28 Important
Misc 29
Misc 30 Important
Misc 31 Important
Misc 32 Important
Misc 33 Important
Misc 34
Misc 35
Misc 36
Misc 37
Misc 38 Important
Misc 39
Misc 40 Important Deleted for CBSE Board 2021 Exams only
Misc 41 Important
Misc 42
Misc 43
Misc 44 Important
Integration Formula Sheet - Chapter 7 Class 12 Formulas Important
About the Author
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.