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

Question

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

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Thiên Thanh 2 months 2021-08-01T05:07:43+00:00 1 Answers 2 views 0

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    2021-08-01T05:08:52+00:00

    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!

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Giải phương trình 1 ẩn: x + 2 - 2(x + 1) = -x . Hỏi x = ? ( )