Evaluate ∫−π$$3$$π log⁡$$($$$$sec$$⁡θ−$$tan$$⁡θ$$)d$$θ .

Question

Evaluate ∫−π$$3$$π log⁡$$($$$$sec$$⁡θ−$$tan$$⁡θ$$)d$$θ .

Infinity Learn · Accepted Answer

Let $$I=$$∫−π$$3$$π log⁡$$($$$$sec$$⁡θ−$$tan$$⁡θ$$)d$$θ$$----------(1)$$ Using the property ∫ab $$f(x)dx=$$∫ab $$f(a+b$$−$$x)dx$$, we get $$I=$$∫−π$$3$$π log⁡[$$sec$$⁡(2$$π−θ$$)−$$tan$$⁡(2$$π−θ$$)]dθ$$=$$∫−π$$3$$π log⁡[$$sec$$⁡θ$$+$$$$tan$$⁡θ]dθ$$------(2)$$ Adding equations (1) and (2), we get $$2I=$$∫−π$$3$$π log⁡[$$($$$$sec$$⁡θ−$$tan$$⁡θ$$)($$$$sec$$⁡θ$$+$$$$tan$$⁡θ$$)$$]dθ   $$=$$∫−π$$3$$π log⁡$$(1)d$$θ$$=$$∫−π$$3$$π $$0$$.dθ$$=0$$⇒$$I=0$$

$Evaluate \int_{- π}^{3 π} \log (\sec θ - \tan θ) d θ .$

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Evaluate ∫−π3π log⁡(sec⁡θ−tan⁡θ)dθ .

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$Evaluate \int_{- π}^{3 π} \log (\sec θ - \tan θ) d θ .$