\int\frac{{x}^{2}}{{x}}\frac{{\left({4}+{9}{x}^{2}\right)}^{1}}{{2}}{\left.{d}{x}\right.}

Cem Hayes 2021-09-06 Answered

\(\int\frac{{x}^{2}}{{x}}\frac{{\left({4}+{9}{x}^{2}\right)}^{1}}{{2}}{\left.{d}{x}\right.}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

ottcomn
Answered 2021-09-07 Author has 23340 answers

Take the integral:
\(\int{x}\sqrt{{{9}{x}^{2}+{4}}}{\left.{d}{x}\right.}\)
For the integrand x \(\sqrt{{{9}{x}^{2}+{4}}}\), substitute \({u}={9}{x}^{2}+{4}{\quad\text{and}\quad}{d}{u}={18}{x}{\left.{d}{x}\right.}\):
\(=\frac{1}{{18}}\int\sqrt{{{u}}}{d}{u}\)
The integral of \(\sqrt{{{u}}}\) is \(\frac{{{2}{u}^{{\frac{3}{{2}}}}}}{{3}}\):
\(=\frac{{u}^{{\frac{3}{{2}}}}}{{27}}+\text{constant}\)
Substitute back for \({u}={9}{x}^{2}+{4}\):
Answer:
\(=\frac{1}{{27}}{\left({9}{x}^{2}+{4}\right)}^{{\frac{3}{{2}}}}+\text{constant}\)

Not exactly what you’re looking for?
Ask My Question
11
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more
...