Definite Integration by properties - P4

Chapter 7 Class 12 Integrals
Concept wise

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### Transcript

Question 2 Evaluate the definite integral β«_0^πβ(π₯ tanβ‘π₯ )/(secβ‘π₯ +γ tanγβ‘π₯ ) ππ₯ Let I=β«_0^πβ(π₯ tanβ‘π₯ )/(secβ‘π₯ +γ tanγβ‘π₯ ) ππ₯ β΄ I=β«_0^πβ((π β π₯) tanβ‘γ (π β π₯)γ)/(secβ‘(π β π₯) +γ tanγβ‘(π β π₯) ) ππ₯ I=β«_0^πβ((π β π₯)(βtanβ‘γ π₯γ) )/((βsecβ‘γ π₯γ) + γ( βtanγβ‘π₯)) ππ₯ I=β«_0^πβ(β(π β π₯) tanβ‘π₯ )/(β(secβ‘π₯ +γ tanγβ‘π₯)) ππ₯ Using The Property, P4 P4 : β«_0^πβγπ(π₯)ππ₯=γ β«_0^πβπ(πβπ₯)ππ₯ I=β«_0^πβ((π β π₯) tanβ‘π₯ )/((secβ‘π₯ +γ tanγβ‘π₯)) ππ₯ Adding (1) and (2) i.e. (1) + (2) I+I=β«_0^πβ(π₯ tanβ‘π₯ )/(secβ‘π₯ +γ tanγβ‘π₯ ) ππ₯+β«_0^πβ(π tanβ‘π₯ β π₯ tanβ‘π₯)/(secβ‘π₯ +γ tanγβ‘π₯ ) ππ₯ 2I=β«_0^πβ(π₯ tanβ‘π₯ + π tanβ‘π₯ β π₯ tanβ‘π₯)/(secβ‘π₯ +γ tanγβ‘π₯ ) ππ₯ 2I=β«_0^πβ(π tanβ‘π₯)/(secβ‘π₯ +γ tanγβ‘π₯ ) ππ₯ 2I=πβ«_0^πβtanβ‘π₯/(secβ‘π₯ +γ tanγβ‘π₯ ) ππ₯ I=π/2 β«_0^πβtanβ‘π₯/(secβ‘π₯ +γ tanγβ‘π₯ ) ππ₯ =π/2 β«_0^πβ(sinβ‘π₯/cosβ‘π₯ )/(1/cosβ‘π₯ + sinβ‘π₯/cosβ‘π₯ ) ππ₯ =π/2 β«_0^πβsinβ‘π₯/(1 + sinβ‘π₯ ) ππ₯ =π/2 β«_0^πβ(sinβ‘π₯ + 1 β 1)/(1 + sinβ‘π₯ ) ππ₯ =π/2 β«_0^πβ[(1 + sinβ‘π₯)/(1 + sinβ‘π₯ ) β1/(1 + sinβ‘π₯ )] ππ₯ =π/2 β«_0^πβ[1 β1/(1 + sinβ‘π₯ )] ππ₯ =π/2 [β«_0^πβ1 ππ₯ββ«_0^πβ1/(1 + sinβ‘π₯ ) ππ₯] =π/2 [[π₯]_0^πββ«_0^πβ1/(1 + sinβ‘π₯ ) ((1 β sinβ‘π₯)/(1 β sinβ‘π₯ )) ππ₯] =π/2 [[πβ0]ββ«_0^πβ(1 β sinβ‘π₯)/(1 β sin^2β‘π₯ ) ππ₯] =π/2 [πββ«_0^πβ(1 β sinβ‘π₯)/cos^2β‘π₯ ππ₯] =π/2 [πββ«_0^πβ[1/cos^2β‘π₯ β sinβ‘π₯/cos^2β‘π₯ ] ππ₯] =π/2 {πββ«_0^πβ[sec^2β‘π₯βtanβ‘π₯ secβ‘π₯ ] ππ₯} =π/2 {πββ«_0^πβsec^2β‘π₯ ππ₯+β«_0^πβγtanβ‘π₯ secβ‘π₯ γ ππ₯} =π/2 [πβ[tanβ‘π₯ ]_0^π+[secβ‘π₯ ]_0^π ] =π/2 {πβ[tanβ‘γ(π)βtanβ‘(0) γ ]+[sec (π)βsecβ‘(0) ]} =π/2 {πβ[0β0]+[β1β1]} =π/2 {πβ0+[β2]} =π/π (πβπ)

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#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.