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 186911 by Spillover last updated on 11/Feb/23

  ∫_0 ^∞ (1/(1+a^x +a^(x/2) ))dx = (1/(ln a))[ln 3−(π/(3(√3)))]

011+ax+ax2dx=1lna[ln3π33]

Answered by witcher3 last updated on 13/Feb/23

a^(x/2) =t,a>1  x=2((ln(t))/(ln(a)))  dx=(2/(ln(a)t))dt  ⇔(2/(ln(a)))∫_1 ^∞ (dt/(t(1+t+t^2 )))  (2/(ln(a)))∫_0 ^1 y(dy/(1+y+y^2 ))  =(2/(ln(a)))∫_0 ^1 ((y+(1/2))/(1+y+y^2 ))−(1/(ln(a)))∫_0 ^1 (dy/((y+(1/2))^2 +(3/4)))  =(2/(ln(a))).(1/2)[ln(1+y+y^2 )]_0 ^1 −(1/(ln(a)))∫_0 ^1 .(4/3)(dy/(1+(((2y)/( (√3)))+(1/( (√3))))))  (1/(ln(a)))[ln(3)−(2/( (√3)))tan^(−1) ((√3))+(2/( (√3)))tan^(−1) ((1/( (√3)))))  =(1/(ln(a)))[ln(3)−(π/( 3(√3)))]

ax2=t,a>1x=2ln(t)ln(a)dx=2ln(a)tdt2ln(a)1dtt(1+t+t2)2ln(a)01ydy1+y+y2=2ln(a)01y+121+y+y21ln(a)01dy(y+12)2+34=2ln(a).12[ln(1+y+y2)]011ln(a)01.43dy1+(2y3+13)1ln(a)[ln(3)23tan1(3)+23tan1(13))=1ln(a)[ln(3)π33]

Terms of Service

Privacy Policy

Contact: info@tinkutara.com