Example 30 - Chapter 7 Class 12 Integrals
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 30 Evaluate β«_0^πβ(π₯ π ππ π₯)/(1 + cos^2β‘π₯ ) ππ₯ Let I=β«_0^πβ(π₯ sinβ‘π₯)/(1 + cos^2β‘π₯ ) ππ₯ β΄ I=β«_0^πβ((π β π₯) π ππ(π β π₯))/(1 + cos^2β‘(π β π₯) ) ππ₯ I=β«_0^πβγ((π β π₯) sinβ‘π₯)/(1 + [βcosβ‘π₯ ]^2 ) ππ₯γ I=β«_0^πβγ(π sinβ‘γπ₯ β π₯ sinβ‘π₯ γ)/(1 + cos^2β‘π₯ ) ππ₯γ Adding (1) and (2) i.e. (1) + (2) I +I=β«_0^πβ(π₯ sinβ‘π₯)/(1 + cos^2β‘π₯ ) ππ₯+β«_0^πβ(π sinβ‘γπ₯ β π₯ π πππ₯γ)/(1 + cos^2β‘π₯ ) ππ₯ 2I =β«_0^πβ(π₯ sinβ‘γπ₯ + π sinβ‘γπ₯ β π₯ sinβ‘π₯ γ γ)/(1 + cos^2β‘π₯ ) ππ₯ 2I =β«_0^πβ(π sinβ‘γπ₯ γ)/(1 + cos^2β‘π₯ ) ππ₯ 2I =πβ«_0^πβsinβ‘γπ₯ γ/(1 + cos^2β‘π₯ ) ππ₯ β΄ I = π/2 β«_0^πβγsinβ‘π₯/(1 + cos^2β‘π₯ ) ππ₯γ Let cos π₯ = t Differentiate both sides w.r.t.π₯ β sinβ‘γπ₯=ππ‘/ππ₯γ ππ₯=ππ‘/(βsinβ‘π₯ ) Putting the values of (cosβ‘π₯ ) and dπ₯, we get I =π/2 β«1_0^πβγsinβ‘π₯/(1 + π‘^2 ).ππ₯γ I = π/2 β«1_0^πβγsinβ‘π₯/(1 + π‘^2 ) Γ ππ‘/(βsinβ‘π₯ )γ I = (βπ)/2 β«1_0^πβγ1/(1 + π‘^2 ) .ππ‘γ I = (βπ)/2 β«1_0^πβγ1/((1)^2 + (π‘)^2 ) .ππ‘γ I=(βπ)/2 [1/1 tan^(β1)β‘(π‘/1) ]_π^π =(βπ)/2 [tan^(β1)β‘(π‘) ]_0^π Putting t = cos x =(βπ)/2 [tan^(β1)β‘(cosβ‘π₯ ) ]_0^π =(βπ)/2 [tan^(β1)β‘γ[cosβ‘π ]βtan^(β1)β‘[cosβ‘0 ] γ ] =(βπ)/2 [tan^(β1)β‘γ(β1)βtan^(β1)β‘(1) γ ] =(βπ)/2 [βtan^(β1)β‘γ(1)βtan^(β1)β‘(1) γ ] =(βπ)/2 [β2 tan^(β1)β‘(1) ] =π[tan^(β1)β‘(1) ] = π[π/4] =π ^π/π
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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo