Ex 7.10, 1 - Chapter 7 Class 12 Integrals
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Ex 7.10, 1 By using the properties of definite integrals, evaluate the integrals : β«_0^(π/2)βγcos^2β‘π₯ ππ₯γ Let π=β«_π^(π /π)βγγπππγ^πβ‘π π πγ I=β«_π^(π /π)βγγππ¨π¬γ^πβ‘ (π /πβπ)π πγ I= β«_π^((π )/π)βγγπ¬π’π§γ^π πγβ‘π π Adding (1) and (2) I+I= β«_0^(π/2)βγcos^2β‘π₯ ππ₯γ + β«_0^((π )/2)βγsin^2 π₯γβ‘ππ₯ 2I= β«_0^((π )/2)β(cos^2β‘γπ₯+sin^2β‘π₯ γ )β‘ππ₯ ππ =β«_π^((π )/π)βγπ .γβ‘π π 2I=[π₯]_0^(π/2) 2I =π/2β0 2I =π/2 π=π /π Evaluate: β«_0^πβγπ^cosβ‘π₯ /(π^cosβ‘π₯ + π^γβcosγβ‘π₯ ) ππ₯γ Let I=β«_0^πβγπ^cosβ‘π₯ /(π^cosβ‘π₯ + π^γβcosγβ‘π₯ ) ππ₯γ " " I= β«_0^πβγπ^cosβ‘γ(π β π₯)γ /(π^cosβ‘γ(π β π₯)γ + π^γβcosγβ‘γ(π β π₯)γ ) ππ₯γ " " I=β«_0^πβγπ^γβcosγβ‘π₯ /(π^γβcosγβ‘π₯ + π^(γβ(βcosγβ‘π₯)) ) ππ₯γ I=β«_0^πβγπ^γβcosγβ‘π₯ /(π^γβcosγβ‘π₯ + π^cosβ‘π₯ ) ππ₯γ Evaluate: β«_0^πβγπ^cosβ‘π₯ /(π^cosβ‘π₯ + π^γβcosγβ‘π₯ ) ππ₯γ Let I=β«_0^πβγπ^cosβ‘π₯ /(π^cosβ‘π₯ + π^γβcosγβ‘π₯ ) ππ₯γ " " I= β«_0^πβγπ^cosβ‘γ(π β π₯)γ /(π^cosβ‘γ(π β π₯)γ + π^γβcosγβ‘γ(π β π₯)γ ) ππ₯γ " " I=β«_0^πβγπ^γβcosγβ‘π₯ /(π^γβcosγβ‘π₯ + π^(γβ(βcosγβ‘π₯)) ) ππ₯γ I=β«_0^πβγπ^γβcosγβ‘π₯ /(π^γβcosγβ‘π₯ + π^cosβ‘π₯ ) ππ₯γ Adding (1) and (2) i.e. (1) + (2) I+I=β«_0^πβγπ^cosβ‘π₯ /(π^cosβ‘π₯ + π^γβcosγβ‘π₯ ) ππ₯γ + β«_0^πβγπ^γβcosγβ‘π₯ /(π^γβcosγβ‘π₯ + π^cosβ‘π₯ ) ππ₯γ 2I=β«_0^πβγ(π^cosβ‘π₯ + π^γβcosγβ‘π₯ )/(π^cosβ‘π₯ + π^γβcosγβ‘π₯ ) ππ₯γ 2I =β«_0^πβγ1 .γβ‘ππ₯ 2I=[π₯]_0^π 2I =πβ0 2I =π π=π /π
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