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Ex 7.11, 7 - Evaluate using properties x (1 - x)n dx - Ex 7.11

Ex 7.11,7 - Chapter 7 Class 12 Integrals - Part 2

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Ex 7.11,7 By using the properties of definite integrals, evaluate the integrals : ∫_0^1β–’γ€– π‘₯(1βˆ’π‘₯)^𝑛 γ€— 𝑑π‘₯ Let I=∫_0^1β–’γ€–π‘₯(1βˆ’π‘₯)^𝑛 𝑑π‘₯γ€— ∴ I=∫_0^1β–’γ€–(1βˆ’π‘₯) [1βˆ’(1βˆ’π‘₯)]^𝑛 𝑑π‘₯γ€— I=∫_0^1β–’γ€–(1βˆ’π‘₯) [1βˆ’1+π‘₯]^𝑛 𝑑π‘₯γ€— I=∫_0^1β–’γ€–(1βˆ’π‘₯) [π‘₯]^𝑛 𝑑π‘₯γ€— I= ∫_0^1β–’γ€–(1βˆ’π‘₯) γ€– π‘₯γ€—^𝑛 𝑑π‘₯γ€— Using P4 : ∫_0^π‘Žβ–’γ€–π‘“(π‘₯)𝑑π‘₯=γ€— ∫_0^π‘Žβ–’π‘“(π‘Žβˆ’π‘₯)𝑑π‘₯ I= ∫_0^1β–’γ€–(γ€– π‘₯γ€—^π‘›βˆ’ π‘₯^(𝑛 + 1) ) 𝑑π‘₯γ€— I= ∫_0^1β–’γ€–γ€– π‘₯γ€—^𝑛 𝑑π‘₯γ€—βˆ’βˆ«_0^1β–’γ€–γ€– π‘₯γ€—^(𝑛 + 1) 𝑑π‘₯γ€— I=[π‘₯^(𝑛 + 1)/(𝑛 + 1)]_0^1βˆ’[π‘₯^(𝑛 + 2)/(𝑛 + 2)]_0^1 I=[(1)^(𝑛 + 1)/(𝑛 + 1)βˆ’(0)^(𝑛 + 1)/(𝑛 + 1)]βˆ’[(1)^(𝑛 + 2)/(𝑛 + 2)βˆ’(0)^(𝑛 + 2)/(𝑛 + 2)] I= 1/(𝑛 + 1)βˆ’1/(𝑛 + 2) I=(𝑛 + 2 βˆ’ (𝑛 + 1))/(𝑛 + 1)(𝑛 + 2) 𝐈=𝟏/(𝒏 + 𝟏)(𝒏 + 𝟐)

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