definite integral?

Question

Hello, does anybody know how to do this?

Integral sign with e on top and L on bottom. (2+ln x)to the power of 3 divide all of that by 5x ??????

thank you in advance.

Answer 1

♣ thus y(x)*dx = 0.2*dx*(2 +ln x)^3 /x;
t = 2 +ln x, x = exp(t-2), dx=dt* exp(t-2);
top limit t2 = 2+ln(e) =3; bottom limit t1 =2+ln(L);
♠ so y(t)*dt = 0.2*dt*t^3, hence
♦ Y(t) = 0.2*t^4 /4 │t= 2+ln(L) ↔ 3│;
Y= 0.05*[3^4 –(2+ln(L))^4];
Looking a bit more decent;

Answer 2

first expand (2+ln x)^3 out to get
8+12ln x +6(ln x)^2 + (ln x)^3
and so your new integral is going to be...

8/5∫1/x+12/5∫ln x/x +6/5∫(ln x)^2/x + 1/5∫(ln x)^3/x
let u = ln x; du = 1/x
8/5∫1/x+12/5∫u du +6/5∫u^2 du + 1/5∫u^3 du

8/5ln|x|+12/5(1/2u^2)+6/5(1/3u^3) + 1/5 (1/4u^4)
8/5ln|x|+12/5(1/2(ln|x|^2)+
6/5(1/3ln|x|^3) + 1/5 (1/4ln|x|^4)
then evaluate at e to L.