"Evaluate the integral. ∫(ln⁡(x))32xdx" solved and explained

Dillard

Answered question

2021-11-06

Evaluate the integral.

\int \frac{{(\ln (x))}^{3}}{2 x} d x

Tasneem Almond

Skilled2021-11-07Added 91 answers

Step 1
Consider the given integral,

\int \frac{{(\ln (x))}^{3}}{2 x} d x

use the substitution,

t = \ln (x)

x = e^{t}

d x = e^{t} d t

so, the integral becomes,

\int \frac{{(\ln (x))}^{3}}{2 x} d x = \int \frac{{(t)}^{3}}{2 e^{t}} e^{t} d t

= \frac{1}{2} \int t^{3} d t

Step 2
now, integrate the above integral,

\int \frac{{(\ln (x))}^{3}}{2 x} d x = \frac{1}{2} \int t^{3} d t

= \frac{1}{2} [\frac{t^{4}}{4}] + c

= \frac{t^{4}}{8} + c

now, undone the substitution,

\int \frac{{(\ln (x))}^{3}}{2 x} d x = \frac{{[\ln (x)]}^{4}}{8} + c

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