Integration

Question

15 / x[In(x)]^4  dx 

 

Ans: -5 / [In(x)]^3 + C

Answer 1

∫15 / x[In(x)]⁴ dx = ?

 

Try a change of variable.

 

Let u(x) = ln(x).

 

du = dx/x ⇒ dx = x du

 

Then

 

∫15 / x[In(x)]⁴ dx = 15 ∫x du / (xu⁴) = 15 ∫du/u⁴ = 15 ∫u^-4 du = 15u^-3/(-3) + C

 

∫15 / x[In(x)]⁴ dx = -5/u³ + C = -5/[ln(x)]³ + C

 

Answer 2

I am very sorry. Please accept my apologies. I completely misread the problem. I had the x term in the denominator to the power 4 as well. That makes the whole thing much more complicated. Upon further research, that integral would boil down to a special function E(t) = integral ( e^(-x)/x) That's two misreadings in one day. Time out.