Evaluate the indefinate integral. \int x^{2}\sin (x^{3})dx

skomminbv 2021-11-22 Answered
Evaluate the indefinate integral.
\(\displaystyle\int{x}^{{{2}}}{\sin{{\left({x}^{{{3}}}\right)}}}{\left.{d}{x}\right.}\)

Expert Community at Your Service

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

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Glenn Cooper
Answered 2021-11-23 Author has 1793 answers
Step 1
Evaluate indefinite integral.
\(\displaystyle\int{x}^{{{2}}}{\sin{{\left({x}^{{{3}}}\right)}}}{\left.{d}{x}\right.}\)
Consider,
\(\displaystyle{u}={x}^{{{3}}}\)
\(\displaystyle{d}{u}={3}{x}^{{{2}}}{\left.{d}{x}\right.}\)
\(\displaystyle{\frac{{{d}{u}}}{{{3}}}}={x}^{{{2}}}{\left.{d}{x}\right.}\)
Step 2
Substitute the values in integral and evaluate.
\(\displaystyle\int{\sin{{\left({u}\right)}}}{\frac{{{d}{u}}}{{{3}}}}\)
\(\displaystyle\Rightarrow{\frac{{{1}}}{{{3}}}}\int{\sin{{u}}}{d}{u}\)
\(\displaystyle\Rightarrow{\frac{{{1}}}{{{3}}}}{\left(-{\cos{{u}}}\right)}\)
Substitute u value,
\(\displaystyle\Rightarrow{\frac{{{1}}}{{{3}}}}{\left(-{\cos{{\left({x}\right)}}}^{{{3}}}\right)}+{C}\)
\(\displaystyle\Rightarrow-{\frac{{{1}}}{{{3}}}}{{\cos{{\left({x}\right)}}}^{{{3}}}+}{C}\)
Have a similar question?
Ask An Expert
0
 
kayleeveez7
Answered 2021-11-24 Author has 250 answers
Step 1: Use Integration by Substitution.
Let \(\displaystyle{u}={x}^{{{3}}},{d}{u}={3}{x}^{{{2}}}{\left.{d}{x}\right.},\ {t}{h}{e}{n}\ {x}^{{{2}}}{\left.{d}{x}\right.}={\frac{{{1}}}{{{3}}}}{d}{u}\)
Step 2: Using u and du above, rewrite \(\displaystyle\int{x}^{{{2}}}{\sin{{\left({x}^{{{3}}}\right)}}}{\left.{d}{x}\right.}\).
\(\displaystyle\int{\frac{{{\sin{{u}}}}}{{{3}}}}{d}{u}\)
Step 3: Use Constant Factor Rule: \(\displaystyle\int{c}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\).
\(\displaystyle{\frac{{{1}}}{{{3}}}}\int{\sin{{u}}}{d}{u}\)
Step 4: Use Trigonometric Integration: the integral of \(\displaystyle{\sin{{u}}}\ {i}{s}\ -{\cos{{u}}}\).
\(\displaystyle-{\frac{{{\cos{{u}}}}}{{{3}}}}\)
Step 5: Substitute \(\displaystyle{u}={x}^{{{3}}}\) back into the original integral.
\(\displaystyle-{\frac{{{\cos{{\left({x}^{{{3}}}\right)}}}}}{{{3}}}}\)
Step 6: Add constant.
\(\displaystyle-{\frac{{{\cos{{\left({x}^{{{3}}}\right)}}}}}{{{3}}}}+{C}\)
0

Expert Community at Your Service

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