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

Evaluate the indefinate integral.
$$\displaystyle\int{x}^{{{2}}}{\sin{{\left({x}^{{{3}}}\right)}}}{\left.{d}{x}\right.}$$

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Glenn Cooper
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,
$$\displaystyle\Rightarrow{\frac{{{1}}}{{{3}}}}{\left(-{\cos{{\left({x}\right)}}}^{{{3}}}\right)}+{C}$$
$$\displaystyle\Rightarrow-{\frac{{{1}}}{{{3}}}}{{\cos{{\left({x}\right)}}}^{{{3}}}+}{C}$$
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kayleeveez7
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.
$$\displaystyle-{\frac{{{\cos{{\left({x}^{{{3}}}\right)}}}}}{{{3}}}}$$
$$\displaystyle-{\frac{{{\cos{{\left({x}^{{{3}}}\right)}}}}}{{{3}}}}+{C}$$