Answer:
[tex] \displaystyle \frac{1}{12} \ln( |{4x}^{ 3} + 9|) + \rm C[/tex]
Step-by-step explanation:
we would like to integrate the following indefinite integral:
[tex] \displaystyle \int \frac{ {x}^{2} }{4 {x}^{3} + 9} dx[/tex]
in order to integrate it we can consider using u-substitution also known as the reverse chain rule and integration by substitution as well
we know that we can use u-substitution if the integral is in the following form
[tex] \displaystyle \int f(g(x))g'(x)dx[/tex]
since our Integral is very close to the form we can use it
let our u and du be 4x³+9 and 12x²dx so that we can transform the Integral
as we don't have 12x² we need a little bit rearrangement
multiply both Integral and integrand by 1/12 and 12:
[tex] \displaystyle \frac{1}{12} \int \frac{ 12{x}^{2} }{4 {x}^{3} + 9} dx[/tex]
apply substitution:
[tex] \displaystyle \frac{1}{12} \int \frac{ 1}{u} du[/tex]
recall Integration rule:
[tex] \displaystyle \frac{1}{12} \ln(|u|) [/tex]
back-substitute:
[tex] \displaystyle \frac{1}{12} \ln( |{4x}^{ 3} + 9|) [/tex]
finally we of course have to add constant of integration:
[tex] \displaystyle \frac{1}{12} \ln( |{4x}^{ 3} + 9|) + \rm C[/tex]
and we are done!
