Question Number 37602 by prof Abdo imad last updated on 15/Jun/18

Commented by prof Abdo imad last updated on 16/Jun/18

Commented by behi83417@gmail.com last updated on 16/Jun/18
![x=a(√(tgt))⇒dx=((a(1+tg^2 t)dt)/(2(√(tgt)))) I=∫((a^2 tgt−1)/((a^4 tg^2 t+a^4 )^2 )).((a(1+tg^2 t)dt)/(2(√(tgt)))) ⇒I=∫((a^2 tgt−1)/(a^8 (tg^2 t+1)^2 )).((a(1+tg^2 t)dt)/(2(√(tgt))))= =(1/a^7 )∫((a^2 tgt−1)/(2(√(tgt))(1+tg^2 t)))dt,tgt=u^2 (1+tg^2 t)dt=2udu⇒dt=((2udu)/(1+u^4 )) ⇒I=(1/a^7 )∫((a^2 u^2 −1)/(2u(1+u^4 ))).((2udu)/(1+u^4 ))=(1/a^7 )∫((a^2 u^2 −1)/((1+u^4 )^2 ))du= =(1/a^7 )[∫((a^2 u^2 )/((1+u^4 )^2 ))du−∫(du/((1+u^4 )^2 ))]=... part#1)∫((u^2 du)/((1+u^4 )^2 ))=(u^3 /(4(u^4 +1)))+((√2)/(16))tg^(−1) ((u(√2))/(1−u^2 ))+ +((√2)/(16))ln((u^2 −u(√2)+1)/( (√(1+u^4 ))))+const part#2)∫(du/((1+u^4 )^2 ))=(u/(4(u^4 +1)))+((3(√2))/(16))tg^(−1) ((u(√2))/(1−u^2 ))− −((3(√2))/(16))ln((u^2 −u(√2)+1)/( (√(1+u^4 ))))+const by symplifing: I=(1/a^5 ).part(#1)−(1/a^7 ).part(#2) change⇒u→(x/a) I=(1/a^5 )[((ax^3 )/(4(a^4 +x^4 )))+((√2)/(16))tg^(−1) (((ax(√2))/(a^2 −x^2 )))− −((3(√2))/(16))ln((x^2 −ax(√2)+a^2 )/( (√(a^4 +x^4 ))))]−(1/a^7 )[((a^3 x)/(4(a^4 +x^4 )))+ +((3(√2))/(16))tg^(−1) ((ax(√2))/(a^2 −x^2 ))−((3(√2))/(16))ln((x^2 −ax(√2)+a^2 )/( (√(a^4 +x^4 ))))]+const 3)I=F(∞)−F(0)=0 4)a=1⇒I=(1/8)[((2x)/(1+x^4 ))+2(√2)tg^(−1) ((x(√2))/(1−x^2 ))− −(√2)ln((x^2 −x(√2)+1)/( (√(1+x^4 ))))]+const .■](https://www.tinkutara.com/question/Q37654.png)
Commented by behi83417@gmail.com last updated on 16/Jun/18

Commented by prof Abdo imad last updated on 16/Jun/18
