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Question Number 30771 by abdo imad last updated on 25/Feb/18

let  I_n = ∫_0 ^(π/4)   (dx/(cos^(2n+1) ))    (n∈N)  1) find a and b fromR /∀x∈[0,(π/4)]  (1/(cosx))=((acosx)/(1−sinx)) +((bcosx)/(1+sinx))  .find  I_0   2) verify the relation  (1/(cos^(2n+3) x))=(1/(cos^(2n+1) x)) +((sinx sinx)/(cos^(2n+3) )) .find the relation  of recurrence between I_n  and I_(n+1)   .

$${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{{cos}^{\mathrm{2}{n}+\mathrm{1}} }\:\:\:\:\left({n}\in{N}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{and}\:{b}\:{fromR}\:/\forall{x}\in\left[\mathrm{0},\frac{\pi}{\mathrm{4}}\right] \\ $$$$\frac{\mathrm{1}}{{cosx}}=\frac{{acosx}}{\mathrm{1}−{sinx}}\:+\frac{{bcosx}}{\mathrm{1}+{sinx}}\:\:.{find}\:\:{I}_{\mathrm{0}} \\ $$$$\left.\mathrm{2}\right)\:{verify}\:{the}\:{relation} \\ $$$$\frac{\mathrm{1}}{{cos}^{\mathrm{2}{n}+\mathrm{3}} {x}}=\frac{\mathrm{1}}{{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}\:+\frac{{sinx}\:{sinx}}{{cos}^{\mathrm{2}{n}+\mathrm{3}} }\:.{find}\:{the}\:{relation} \\ $$$${of}\:{recurrence}\:{between}\:{I}_{{n}} \:{and}\:{I}_{{n}+\mathrm{1}} \:\:. \\ $$

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