What is the indefinite integral of # ((lnx)^2)/x#?

2 Answers
Jun 27, 2016

#((ln x)^3)/3+C#

Explanation:

Do a substitution: #u=ln x#, #du = 1/x dx# to get:

#\int\ ((ln(x))^2)/x\ dx=\int\ u^{2}\ du=u^{3}/3+C=((ln x)^3)/3+C#

Jun 27, 2016

# =1/3 (ln x)^3 + C#

Explanation:

#int \((lnx)^2)/x \ dx#

another integration where knowing the commonest calculus patterns comes in very useful

so we have the pattern that #d/dx (f(x))^3 = 3 (f(x) )^2f'(x)#

and #d/dx ln (g(x)) = 1/(g(x)) g'(x)#

combining these ideas, watch what happens if we do

#d/dx (ln x)^3#

#= color{red}{3} (ln x)^2 * 1/x = color{red}{3} ((ln x)^2)/x#

so we do #color{red}{1/3} d/dx (ln x)^3 = d/dx (color{red}{1/3}(ln x)^3) = ((ln x)^2)/x#

if follows that

# int \ ((ln x)^2)/x \ dx =1/3 (ln x)^3 + C#