Evaluate the integral. \int x^{3} \ln x dx

Evaluate the integral. \int x^{3} \ln x dx

Question
Applications of integrals
asked 2020-10-19
Evaluate the integral.
\(\displaystyle\int{x}^{{{3}}}{\ln{{x}}}{\left.{d}{x}\right.}\)

Answers (1)

2020-10-20
Apply integration by parts:
\(\displaystyle\int{u}{d}{v}={u}{v}-\int{v}{d}{u}\)
Let, \(\displaystyle{u}={\ln{{x}}}\)
\(\displaystyle{v}={\frac{{{x}^{{{4}}}}}{{{4}}}}\)
\(\displaystyle{v}'={x}^{{{3}}}\)
\(\displaystyle{d}{v}={x}^{{{3}}}{\left.{d}{x}\right.}\)
\(\displaystyle\int{x}^{{{3}}}{\ln{{x}}}{\left.{d}{x}\right.}={\frac{{{1}}}{{{4}}}}{x}^{{{4}}}{\ln{{\left({x}\right)}}}-\int{\frac{{{x}^{{{3}}}}}{{{4}}}}{\left.{d}{x}\right.}\)
\(\displaystyle={\frac{{{1}}}{{{4}}}}{x}^{{{4}}}{\ln{{\left({x}\right)}}}-{\frac{{{x}^{{{4}}}}}{{{16}}}}+{C}\)
0

Relevant Questions

asked 2021-02-08
Evaluate the integral.
\(\displaystyle\int{x}^{{{2}}}{\ln{{x}}}{\left.{d}{x}\right.}\)
asked 2021-03-09
Evaluate the following integral.
\(\displaystyle\int{2}{x}{\left({1}-{x}^{{-{3}}}\right)}{\left.{d}{x}\right.}\)
asked 2020-11-05
Evaluate the following integral.
\(\displaystyle\int{x}^{{-{3}}}{\left({x}+{1}\right)}{\left.{d}{x}\right.}\)
asked 2021-03-07
Evaluate the integral.
\(\displaystyle\int\frac{{{3}-{x}}}{{\sqrt{{x}}}}{\left.{d}{x}\right.}\)
asked 2020-10-21
Evaluate the integral.
\(\displaystyle\int\sqrt{{x}}{\left({x}^{{3}}+\frac{{x}}{{2}}\right)}{\left.{d}{x}\right.}\)
asked 2021-01-15
Evaluate the integral.
\(\displaystyle\int{\left(\frac{{2}}{{x}^{{3}}}+\frac{{1}}{\sqrt{{x}}}\right)}{\left.{d}{x}\right.}\)
asked 2021-02-24
Solve the integral.
\(\displaystyle\int{x}{\ln{{x}}}{\left.{d}{x}\right.}\)
asked 2021-02-14
Evaluate the following integral.
\(\displaystyle\int{\frac{{{x}}}{{\sqrt{{{x}-{4}}}}}}{\left.{d}{x}\right.}\)
asked 2020-11-09
Evaluate the integral.
\(\displaystyle\int\frac{{{x}^{{2}}-{1}}}{{{x}^{{2}}}}{\left.{d}{x}\right.}\)
asked 2020-11-08
Evaluate the integral \(\displaystyle\int{\frac{{{\cos{{\left({1}-{\ln{{\left({y}\right)}}}\right)}}}}}{{{y}}}}{\left.{d}{y}\right.}\)
...