Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 176679 by mnjuly1970 last updated on 24/Sep/22

Eeasy integral.... 𝛀 = ∫_(βˆ’βˆ«_0 ^( ∞) e^( βˆ’x^( 2) ) dx) ^( ∫_0 ^( ∞) e^( βˆ’x^( 2) ) dx) sin^( 2) (t).ln^( 3) ( t + (√(1+t^( 2) )))dt βˆ’βˆ’βˆ’m.nβˆ’βˆ’βˆ’

$$ \\ $$$$\:\:\:\:\:\:\:\:{Eeasy}\:\:{integral}.... \\ $$$$\:\:\:\:\:\:\boldsymbol{\Omega}\:=\:\int_{βˆ’\int_{\mathrm{0}} ^{\:\infty} {e}^{\:βˆ’{x}^{\:\mathrm{2}} } {dx}} ^{\:\int_{\mathrm{0}} ^{\:\infty} {e}^{\:βˆ’{x}^{\:\mathrm{2}} } {dx}} {sin}^{\:\mathrm{2}} \left({t}\right).{ln}^{\:\mathrm{3}} \left(\:{t}\:+\:\sqrt{\mathrm{1}+{t}^{\:\mathrm{2}} }\right){dt}\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:βˆ’βˆ’βˆ’{m}.{n}βˆ’βˆ’βˆ’ \\ $$

Answered by Peace last updated on 24/Sep/22

ln(βˆ’t+(√(1+t^2 )))=βˆ’ln(t+(√(1+t^2 ))) Ξ©=0

$${ln}\left(βˆ’{t}+\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\right)=βˆ’{ln}\left({t}+\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\right) \\ $$$$\Omega=\mathrm{0} \\ $$

Answered by mahdipoor last updated on 25/Sep/22

βˆ’f(a)=βˆ’[sin^2 (a).ln^3 (a+(√(a^2 +1)))]= sin^2 (a).[βˆ’ln(a+(√(a^2 +1)))]^3 = sin^2 (a).ln^3 ((1/(a+(√(a^2 +1)))))= sin^2 (a).ln^3 (((βˆ’a+(√(a^2 +1)))/([a+(√(a^2 +1))][βˆ’a+(√(a^2 +1))])))= sin^2 (βˆ’a).ln^3 (βˆ’a+(√((βˆ’a)^2 +1)))=f(βˆ’a) β‡’β‡’ βˆ€a , f(ΞΎ).Ξ΄x+f(βˆ’ΞΎ).Ξ΄x=0 β‡’β‡’ get ∫_0 ^∞ e^(βˆ’x^2 ) dx=m ΞΎ_i =i.Ξ΄x β‡’β‡’ 0=lim_(Ξ΄xβ†’0) Ξ£_(i=0) ^(m/Ξ΄x) f(βˆ’ΞΎ_i ).Ξ΄x+f(ΞΎ_i ).Ξ΄x= lim_(Ξ΄xβ†’0) Ξ£_(βˆ’m/Ξ΄x) ^(m/Ξ΄x) f(ΞΎ_i ).Ξ΄x=∫_( βˆ’m) ^( m) f(x)dx

$$βˆ’{f}\left({a}\right)=βˆ’\left[{sin}^{\mathrm{2}} \left({a}\right).{ln}^{\mathrm{3}} \left({a}+\sqrt{{a}^{\mathrm{2}} +\mathrm{1}}\right)\right]= \\ $$$${sin}^{\mathrm{2}} \left({a}\right).\left[βˆ’{ln}\left({a}+\sqrt{{a}^{\mathrm{2}} +\mathrm{1}}\right)\right]^{\mathrm{3}} = \\ $$$${sin}^{\mathrm{2}} \left({a}\right).{ln}^{\mathrm{3}} \left(\frac{\mathrm{1}}{{a}+\sqrt{{a}^{\mathrm{2}} +\mathrm{1}}}\right)= \\ $$$${sin}^{\mathrm{2}} \left({a}\right).{ln}^{\mathrm{3}} \left(\frac{βˆ’{a}+\sqrt{{a}^{\mathrm{2}} +\mathrm{1}}}{\left[{a}+\sqrt{{a}^{\mathrm{2}} +\mathrm{1}}\right]\left[βˆ’{a}+\sqrt{{a}^{\mathrm{2}} +\mathrm{1}}\right]}\right)= \\ $$$${sin}^{\mathrm{2}} \left(βˆ’{a}\right).{ln}^{\mathrm{3}} \left(βˆ’{a}+\sqrt{\left(βˆ’{a}\right)^{\mathrm{2}} +\mathrm{1}}\right)={f}\left(βˆ’{a}\right) \\ $$$$\Rightarrow\Rightarrow \\ $$$$\forall{a}\:,\:\:{f}\left(\xi\right).\delta{x}+{f}\left(βˆ’\xi\right).\delta{x}=\mathrm{0} \\ $$$$\Rightarrow\Rightarrow \\ $$$${get}\:\int_{\mathrm{0}} ^{\infty} {e}^{βˆ’{x}^{\mathrm{2}} } {dx}={m} \\ $$$$\xi_{{i}} ={i}.\delta{x} \\ $$$$\Rightarrow\Rightarrow \\ $$$$\mathrm{0}=\underset{\delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\underset{{i}=\mathrm{0}} {\overset{{m}/\delta{x}} {\sum}}{f}\left(βˆ’\xi_{{i}} \right).\delta{x}+{f}\left(\xi_{{i}} \right).\delta{x}= \\ $$$$\underset{\delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\underset{βˆ’{m}/\delta{x}} {\overset{{m}/\delta{x}} {\sum}}{f}\left(\xi_{{i}} \right).\delta{x}=\int_{\:βˆ’{m}} ^{\:{m}} {f}\left({x}\right){dx} \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com