Misc 35 - Chapter 7 Class 12 Integrals
Last updated at April 16, 2024 by Teachoo
Miscellaneous
Misc 1
Important
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Chapter 7 Class 12 Integrals
Serial order wise
Transcript
Misc 35 Prove that β«_0^(π/2)βsin^3β‘π₯ ππ₯=2/3 Solving L.H.S β«_0^(π/2)βsin^3β‘π₯ ππ₯ = β«_0^(π/2)βγ γsin π₯ (sinγ^2β‘γπ₯)γ γ ππ₯ = β«_0^(π/2)βπ ππβ‘γπ₯ (1βγπππ γ^2 π₯)γ ππ₯ = β«_0^(π/2)βπ ππβ‘γπ₯ ππ₯β β«1_0^(π/2)βγsinβ‘γπ₯ γπππ γ^2 π₯γ ππ₯γγ π°_π β«1_0^(π/2)βsinβ‘γπ₯ ππ₯γ = β [cosβ‘π₯ ]_0^(π/2) = β[0β1] = 1 π°_π β«1_0^(π/2)βsinβ‘γπ₯ γπππ γ^2 π₯ ππ₯γ Let t = cos x ππ‘/ππ₯ = - sin x dt = β sin x dx Substituting, β«1_0^1βsinβ‘γπ₯Γπ‘^2Γγ ππ‘/(βsinβ‘γπ₯ γ ) = ββ«1_1^0βγπ‘^2 ππ‘γ = γβ[π‘^3/3]γ_1^0 = β("0 β " 1/3)=β((β1)/3) = 1/3 L.H.S = πΌ_1β πΌ_2 = 1 β 1/3 = π/π = R.H.S Hence, proved.
