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Question Number 150372 by tabata last updated on 11/Aug/21

Commented by mathdanisur last updated on 11/Aug/21

∫_a ^(a+1) f(x)dx=2  ⇒∫_a ^(a+1) ((1/(1+x)) + (1/(1+x^2 )) + (1/(1+x^3 )) + ... + (1/(1+x^n )))dx = 2  (1)  ∫_a ^(a+1) f((1/x))dx  ⇒∫_a ^(a+1) ((x/(1+x)) + (x^2 /(1+x^2 )) + (x^3 /(1+x^3 )) + ... + (x^n /(1+x^n )))dx = z  (2)  ⇒ (1) + (2)  ⇒ ∫_a ^(a+1) (n) dx = 2 + z ⇒ z = n - 2 ▲  ⇒ ∫_a ^(a+1) f ((1/x)) dx = n - 2 ▲

$$\underset{{a}} {\overset{{a}+\mathrm{1}} {\int}}{f}\left({x}\right){dx}=\mathrm{2}\:\:\Rightarrow\underset{{a}} {\overset{{a}+\mathrm{1}} {\int}}\left(\frac{\mathrm{1}}{\mathrm{1}+{x}}\:+\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{3}} }\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{{n}} }\right){dx}\:=\:\mathrm{2}\:\:\left(\mathrm{1}\right) \\ $$$$\underset{{a}} {\overset{{a}+\mathrm{1}} {\int}}{f}\left(\frac{\mathrm{1}}{{x}}\right){dx}\:\:\Rightarrow\underset{{a}} {\overset{{a}+\mathrm{1}} {\int}}\left(\frac{{x}}{\mathrm{1}+{x}}\:+\:\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:+\:\frac{{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{3}} }\:+\:...\:+\:\frac{{x}^{{n}} }{\mathrm{1}+{x}^{{n}} }\right){dx}\:=\:\boldsymbol{{z}}\:\:\left(\mathrm{2}\right) \\ $$$$\Rightarrow\:\left(\mathrm{1}\right)\:+\:\left(\mathrm{2}\right)\:\:\Rightarrow\:\underset{{a}} {\overset{{a}+\mathrm{1}} {\int}}\left({n}\right)\:{dx}\:=\:\mathrm{2}\:+\:\boldsymbol{{z}}\:\Rightarrow\:\boldsymbol{{z}}\:=\:{n}\:-\:\mathrm{2}\:\blacktriangle \\ $$$$\Rightarrow\:\underset{{a}} {\overset{{a}+\mathrm{1}} {\int}}{f}\:\left(\frac{\mathrm{1}}{{x}}\right)\:{dx}\:=\:{n}\:-\:\mathrm{2}\:\blacktriangle \\ $$

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