Question

Workout to the simplest:5E%7B3%7D%20%29%20dx" id="TexFormula1" title=" \int \: {x}^{2} ln( {x}^{3} ) dx" alt=" \int \: {x}^{2} ln( {x}^{3} ) dx" align="absmiddle" class="latex-formula">
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Answer 1

Answer:

$\rm \displaystyle \ln(x) { {x}^{3} } - \frac{ {x}^{3} }{3} + \rm C$

Step-by-step explanation:

we would like to integrate the following integration

$\displaystyle \int {x}^{2} \ln( {x}^{3} ) dx$

before doing so we can use logarithm exponent rule in order to get rid of the exponent of ln(x³)

$\displaystyle \int 3 {x}^{2} \ln( {x}^{} ) dx$

now notice that the integrand is in the mutilation of two different functions thus we can use integration by parts given by

$\rm\displaystyle \int u \cdot \: vdx = u \int vdx - \int u' \bigg( \int vdx \bigg)dx$

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

$\displaystyle u = \ln(x) \quad \text{and} \quad v = {3x}^{2}$

and u' is $\frac{1}{x}$

altogether substitute:

$\rm \displaystyle \ln(x) \int 3{x}^{2} dx - \int \frac{1}{x} \left( \int 3 {x}^{2} dx \right)dx$

use exponent integration rule to integrate exponent:

$\rm \displaystyle \ln(x) \int 3{x}^{2} dx - \int \frac{1}{x} \left( 3\frac{ {x}^{3} }{3} \right)dx$

once again exponent integration rule:

$\rm \displaystyle \ln(x) 3\frac{ {x}^{3} }{3} - \int \frac{1}{x} \left( 3\frac{ {x}^{3} }{3} \right)dx$

simplify integrand:

$\rm \displaystyle \ln(x) 3\frac{ {x}^{3} }{3} - \int \frac{ 3{x}^{3} }{3x} dx$

use law of exponent to simplify exponent:

$\rm \displaystyle \ln(x) \frac{ 3{x}^{3} }{3} - \int \frac{ 3\cancel{ {x}^{3}} }{3 \cancel{x}} dx$

$\rm \displaystyle \ln(x) \frac{ 3{x}^{3} }{3} - \int \frac{ 3{x}^{3} }{3} dx$

use constant integration rule to get rid of constant:

$\rm \displaystyle \ln(x) \frac{3 {x}^{3} }{3} - 1 \int {x}^{2}dx$

use exponent integration rule:

$\rm \displaystyle \ln(x) \frac{3 {x}^{3} }{3} - \frac{ {x}^{3} }{3}$

$\rm \displaystyle \ln(x) { {x}^{3} } - \frac{ {x}^{3} }{3}$

and finally we of course have to add the constant of integration:

$\rm \displaystyle \ln(x) { {x}^{3} } - \frac{ {x}^{3} }{3} + \rm C$

and we are done!

Answer 2

Answer:

Step-by-step explanation: