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Ex 7.11, 11 - Evaluate definite integral sin2 x dx - Definate Integration by properties - P7

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  1. Chapter 7 Class 12 Integrals
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Ex 7.11, 11 By using the properties of definite integrals, evaluate the integrals : ∫_((βˆ’ πœ‹)/2)^(πœ‹/2)β–’γ€– sin^2〗⁑π‘₯ 𝑑π‘₯ This is of form ∫_(βˆ’π‘Ž)^π‘Žβ–’π‘“(π‘₯)𝑑π‘₯ 𝑓(π‘₯)=sin^2⁑π‘₯ 𝑓(βˆ’π‘₯)=sin^2⁑(βˆ’π‘₯)=(βˆ’π‘ π‘–π‘›π‘₯)^2=sin^2⁑π‘₯ ∴ 𝑓(π‘₯)=𝑓(βˆ’π‘₯) ∴ ∫_((βˆ’πœ‹)/2)^(πœ‹/2)β–’γ€–sin^2⁑〖π‘₯ 𝑑π‘₯γ€—=2∫_0^(πœ‹/2)β–’γ€–sin^2 π‘₯ 𝑑π‘₯γ€—γ€— =2∫_0^(πœ‹/2)β–’[(1 βˆ’ cos⁑2π‘₯)/2]𝑑π‘₯ =∫_0^(πœ‹/2)β–’γ€–(1βˆ’cos⁑2π‘₯ ) 𝑑π‘₯γ€— = [π‘₯ βˆ’sin⁑2π‘₯/2]_0^(πœ‹/2) = πœ‹/2βˆ’sin⁑2(πœ‹/2)/2βˆ’0+sin⁑〖2(0)γ€—/2 = πœ‹/2βˆ’sinβ‘πœ‹/2+sin⁑0/2 = πœ‹/2βˆ’0+0 = πœ‹/2

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