Menu Close

1-n-x-x-2-dx-




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

Leave a Reply

Your email address will not be published. Required fields are marked *