Definite Integration by properties - P7

Chapter 7 Class 12 Integrals
Concept wise

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### Transcript

Ex 7.11, 11 By using the properties of definite integrals, evaluate the integrals : β«_((β π)/2)^(π/2)βγ sin^2γβ‘π₯ ππ₯ This is of form β«_(βπ)^πβπ(π₯)ππ₯ where π(π₯)=sin^2β‘π₯ π(βπ₯)=sin^2β‘(βπ₯)=(βπ πππ₯)^2=sin^2β‘π₯ β΄ π(π₯)=π(βπ₯) Using the Property β«_(βπ)^πβγπ(π₯)ππ₯=2,γ β«_0^πβγπ(π₯)ππ₯ γ if f(βπ₯)=π(π₯) β΄ β«_((βπ)/2)^(π/2)βγsin^2β‘γπ₯ ππ₯γ=2β«_0^(π/2)βγγπππγ^π π ππ₯γγ =2β«_0^(π/2)β[(π β πππβ‘ππ)/π]ππ₯ =β«_0^(π/2)βγ(1βcosβ‘2π₯ ) ππ₯γ = [π₯ βsinβ‘2π₯/2]_0^(π/2) = [π/2βsinβ‘2(π/2)/2]β [0βsinβ‘γ2(0)γ/2] = π/2βsinβ‘π/2β0 = π/2β0+0 = π/π β΅ cos 2x = 1 β 2 γπ ππγ^2 π₯ β 2 γπ ππγ^2 π₯ = 1 β cos 2x β γπ ππγ^2 π₯ = "1 β cos 2x" /2