Question

Find the indefinite integral ∫((x^2)/(4x^3+9)) dx

Answer 1

Answer:

$\displaystyle \frac{1}{12} \ln( |{4x}^{ 3} + 9|) + \rm C$

Step-by-step explanation:

we would like to integrate the following indefinite integral:

$\displaystyle \int \frac{ {x}^{2} }{4 {x}^{3} + 9} dx$

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

$\displaystyle \int f(g(x))g'(x)dx$

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:

$\displaystyle \frac{1}{12} \int \frac{ 12{x}^{2} }{4 {x}^{3} + 9} dx$

apply substitution:

$\displaystyle \frac{1}{12} \int \frac{ 1}{u} du$

recall Integration rule:

$\displaystyle \frac{1}{12} \ln(|u|)$

back-substitute:

$\displaystyle \frac{1}{12} \ln( |{4x}^{ 3} + 9|)$

finally we of course have to add constant of integration:

$\displaystyle \frac{1}{12} \ln( |{4x}^{ 3} + 9|) + \rm C$

and we are done!