Answer:
[tex]\rm \displaystyle \ln(x) { {x}^{3} } - \frac{ {x}^{3} }{3} + \rm C[/tex]
Step-by-step explanation:
we would like to integrate the following integration
[tex] \displaystyle \int {x}^{2} \ln( {x}^{3} ) dx[/tex]
before doing so we can use logarithm exponent rule in order to get rid of the exponent of ln(x³)
[tex] \displaystyle \int 3 {x}^{2} \ln( {x}^{} ) dx[/tex]
now notice that the integrand is in the mutilation of two different functions thus we can use integration by parts given by
[tex] \rm\displaystyle \int u \cdot \: vdx = u \int vdx - \int u' \bigg( \int vdx \bigg)dx[/tex]
where u' can be defined by the differentiation of u
first we need to choose our u and v in that case we'll choose u which comes first in the guideline ILATE which full from is Inverse trig, Logarithm, Algebraic expression, Trigonometry, Exponent.
since Logarithms come first our
[tex] \displaystyle u = \ln(x) \quad \text{and} \quad v = {3x}^{2} [/tex]
and u' is [tex]\frac{1}{x}[/tex]
altogether substitute:
[tex] \rm \displaystyle \ln(x) \int 3{x}^{2} dx - \int \frac{1}{x} \left( \int 3 {x}^{2} dx \right)dx[/tex]
use exponent integration rule to integrate exponent:
[tex] \rm \displaystyle \ln(x) \int 3{x}^{2} dx - \int \frac{1}{x} \left( 3\frac{ {x}^{3} }{3} \right)dx[/tex]
once again exponent integration rule:
[tex] \rm \displaystyle \ln(x) 3\frac{ {x}^{3} }{3} - \int \frac{1}{x} \left( 3\frac{ {x}^{3} }{3} \right)dx[/tex]
simplify integrand:
[tex] \rm \displaystyle \ln(x) 3\frac{ {x}^{3} }{3} - \int \frac{ 3{x}^{3} }{3x} dx[/tex]
use law of exponent to simplify exponent:
[tex] \rm \displaystyle \ln(x) \frac{ 3{x}^{3} }{3} - \int \frac{ 3\cancel{ {x}^{3}} }{3 \cancel{x}} dx[/tex]
[tex] \rm \displaystyle \ln(x) \frac{ 3{x}^{3} }{3} - \int \frac{ 3{x}^{3} }{3} dx[/tex]
use constant integration rule to get rid of constant:
[tex]\rm \displaystyle \ln(x) \frac{3 {x}^{3} }{3} - 1 \int {x}^{2}dx[/tex]
use exponent integration rule:
[tex]\rm \displaystyle \ln(x) \frac{3 {x}^{3} }{3} - \frac{ {x}^{3} }{3} [/tex]
[tex]\rm \displaystyle \ln(x) { {x}^{3} } - \frac{ {x}^{3} }{3} [/tex]
and finally we of course have to add the constant of integration:
[tex]\rm \displaystyle \ln(x) { {x}^{3} } - \frac{ {x}^{3} }{3} + \rm C[/tex]
and we are done!
