Menu Close

Question-48823




Question Number 48823 by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18
Commented by Abdulhafeez Abu qatada last updated on 29/Nov/18
This intergrals i must try
$${This}\:{intergrals}\:{i}\:{must}\:{try} \\ $$
Commented by Meritguide1234 last updated on 30/Nov/18
Commented by tanmay.chaudhury50@gmail.com last updated on 30/Nov/18
excellent sir...∫_a ^b f(x)dx=∫_a ^b f(a+b−x)dx
$${excellent}\:{sir}…\int_{{a}} ^{{b}} {f}\left({x}\right){dx}=\int_{{a}} ^{{b}} {f}\left({a}+{b}−{x}\right){dx} \\ $$$$ \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 01/Dec/18
7)(1/(1+x^2 ))<(1/x^2 )  ∫_0 ^1 (1/((1+x^2 )^n ))dx<∫_0 ^1 (1/x^(2n) )dx  I<∣(x^(−2n+1) /(−2n+1))∣_0 ^1   I<(1/(1−2n))(1−0)  so lim_(n→∞) (√n) ∫_0 ^1 (dx/((1+x^2 )^n ))<lim_(n→∞) (√n) ∫_0 ^1 (dx/x^(2n) )  I_(question) <lim_(n→∞) (√n) ×(1/(1−2n))  I_(question) <lim_(n→∞) (√n) ×((1/n)/(((1/n)−2)))=lim_(n→∞) (1/( (√n)))×(1/(((1/n)−2)))=0  pls check...
$$\left.\mathrm{7}\right)\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }<\frac{\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }{dx}<\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{{x}^{\mathrm{2}{n}} }{dx} \\ $$$${I}<\mid\frac{{x}^{−\mathrm{2}{n}+\mathrm{1}} }{−\mathrm{2}{n}+\mathrm{1}}\mid_{\mathrm{0}} ^{\mathrm{1}} \\ $$$${I}<\frac{\mathrm{1}}{\mathrm{1}−\mathrm{2}{n}}\left(\mathrm{1}−\mathrm{0}\right) \\ $$$${so}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{{n}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }<\underset{{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{{n}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dx}}{{x}^{\mathrm{2}{n}} } \\ $$$${I}_{{question}} <\underset{{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{{n}}\:×\frac{\mathrm{1}}{\mathrm{1}−\mathrm{2}{n}} \\ $$$${I}_{{question}} <\underset{{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{{n}}\:×\frac{\frac{\mathrm{1}}{{n}}}{\left(\frac{\mathrm{1}}{{n}}−\mathrm{2}\right)}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\:\sqrt{{n}}}×\frac{\mathrm{1}}{\left(\frac{\mathrm{1}}{{n}}−\mathrm{2}\right)}=\mathrm{0} \\ $$$${pls}\:{check}… \\ $$$$ \\ $$

Leave a Reply

Your email address will not be published.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