Question and Answers Forum

All Questions      Topic List

Relation and Functions Questions

Previous in All Question      Next in All Question      

Previous in Relation and Functions      Next in Relation and Functions      

Question Number 180901 by alcohol last updated on 19/Nov/22

∫_1 ^( n) ((⌊x⌋)/x^2 )dx =

$$\int_{\mathrm{1}} ^{\:{n}} \frac{\lfloor{x}\rfloor}{{x}^{\mathrm{2}} }{dx}\:=\: \\ $$

Answered by srikanth2684 last updated on 19/Nov/22

∫_1 ^( 2) ((⌊x⌋)/x^2 )dx +∫_2 ^( 3) ((⌊x⌋)/x^2 )dx +  ∫_3 ^( 4) ((⌊x⌋)/x^2 )dx +...∫_(n−1) ^( n) ((⌊x⌋)/x^2 )dx   =∫_1 ^( 2) (1/x^2 )dx +∫_2 ^( 3) (2/x^2 )dx +  ∫_3 ^( 4) (3/x^2 )dx +...∫_(n−1) ^( n) ((n−1)/x^2 )dx   =(((−1)/x))_1 ^2 +2(((−1)/x))_2 ^3 +  3(((−1)/x))_3 ^4 +...+(n−1)(((−1)/x))_(n−1) ^n   =(1−(1/2))+2((1/2)−(1/3))+  ...+(n−1)((1/(n−1))−(1/n))  =(1/2)+(1/3)+...+(1/n)

$$\int_{\mathrm{1}} ^{\:\mathrm{2}} \frac{\lfloor{x}\rfloor}{{x}^{\mathrm{2}} }{dx}\:+\int_{\mathrm{2}} ^{\:\mathrm{3}} \frac{\lfloor{x}\rfloor}{{x}^{\mathrm{2}} }{dx}\:+ \\ $$$$\int_{\mathrm{3}} ^{\:\mathrm{4}} \frac{\lfloor{x}\rfloor}{{x}^{\mathrm{2}} }{dx}\:+...\int_{{n}−\mathrm{1}} ^{\:{n}} \frac{\lfloor{x}\rfloor}{{x}^{\mathrm{2}} }{dx}\: \\ $$$$=\int_{\mathrm{1}} ^{\:\mathrm{2}} \frac{\mathrm{1}}{{x}^{\mathrm{2}} }{dx}\:+\int_{\mathrm{2}} ^{\:\mathrm{3}} \frac{\mathrm{2}}{{x}^{\mathrm{2}} }{dx}\:+ \\ $$$$\int_{\mathrm{3}} ^{\:\mathrm{4}} \frac{\mathrm{3}}{{x}^{\mathrm{2}} }{dx}\:+...\int_{{n}−\mathrm{1}} ^{\:{n}} \frac{{n}−\mathrm{1}}{{x}^{\mathrm{2}} }{dx}\: \\ $$$$=\left(\frac{−\mathrm{1}}{{x}}\right)_{\mathrm{1}} ^{\mathrm{2}} +\mathrm{2}\left(\frac{−\mathrm{1}}{{x}}\right)_{\mathrm{2}} ^{\mathrm{3}} + \\ $$$$\mathrm{3}\left(\frac{−\mathrm{1}}{{x}}\right)_{\mathrm{3}} ^{\mathrm{4}} +...+\left({n}−\mathrm{1}\right)\left(\frac{−\mathrm{1}}{{x}}\right)_{{n}−\mathrm{1}} ^{{n}} \\ $$$$=\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right)+\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}}\right)+ \\ $$$$...+\left({n}−\mathrm{1}\right)\left(\frac{\mathrm{1}}{{n}−\mathrm{1}}−\frac{\mathrm{1}}{{n}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+...+\frac{\mathrm{1}}{{n}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com