$\int x^2\sin \left(x\right) \mathit{\ⅆ}x$

$\=x^2\left(-\cos \left(x\right)\right)-\int 2 x \left(-\cos \left(x\right)\right) \ⅆx$

$u\=x^2\Rightarrow \mathrm{du}\=2 x \mathrm{dx}$

$\mathrm{dv}\=\sin \left(x\right)\mathrm{dx}\Rightarrow v\=-\cos \left(x\right)$

$\=-x^2\cos \left(x\right)\+2\int x \cos \left(x\right) \ⅆx$

$\=-x^2\cos \left(x\right)\+2\left(x \sin \left(x\right)-\int \sin \left(x\right) \ⅆx\right)$

$u\=x\Rightarrow \mathrm{du}\=\mathrm{dx}$ 

$\mathrm{dv}\=\cos \left(x\right)\mathrm{dx}\Rightarrow v\=\sin \left(x\right)$

$\=-x^2\cos \left(x\right)\+2\left(x \sin \left(x\right)-\left(- \cos \left(x\right)\right)\right)$

$\=-x^2\cos \left(x\right)\+2 x \sin \left(x\right)\+2 \cos \left(x\right)$

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$\int x^2\sin \left(x\right) \mathit{\ⅆ}x$$\stackrel{\text{show solution steps}}{\to }$$\begin{array}{lll}& & \text{Integration Steps}\\ & & \int x^2\sin \left(x\right)\ⅆx\\ \text{▫}& & \text{1. Apply integration by Parts}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{Parts}}\text{rule}\\ & & \int u\ⅆv=vu-\left(\int v\ⅆu\right)\\ & \text{◦}& \text{First part}\\ & & u=x^2\\ & \text{◦}& \text{Second part}\\ & & \mathrm{dv}=\sin \left(x\right)\\ & \text{◦}& \text{Differentiate first part}\\ & & \mathrm{du}=\frac{\ⅆ}{\ⅆx}\left(x^2\right)\\ & & \mathrm{du}=2x\\ & \text{◦}& \text{Integrate second part}\\ & & v=\int \sin \left(x\right)\ⅆx\\ & & v=-\cos \left(x\right)\\ & & \int x^2\sin \left(x\right)\ⅆx=-x^2\cos \left(x\right)-\left(\int -2\cos \left(x\right)x\ⅆx\right)\\ & & \text{This gives:}\\ & & -x^2\cos \left(x\right)-\left(\int -2\cos \left(x\right)x\ⅆx\right)\\ \text{▫}& & \text{2. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\int -2\cos \left(x\right)x\ⅆx\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant multiple}}\text{rule}\\ & & \int Cf\left(x\right)\ⅆx=C\left(\int f\left(x\right)\ⅆx\right)\\ & \text{◦}& \text{This means:}\\ & & \int -2\cos \left(x\right)x\ⅆx=-2\left(\int \cos \left(x\right)x\ⅆx\right)\\ & & \text{We can rewrite the integral as:}\\ & & -x^2\cos \left(x\right)+2\left(\int \cos \left(x\right)x\ⅆx\right)\\ \text{▫}& & \text{3. Apply integration by Parts}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{Parts}}\text{rule}\\ & & \int u\ⅆv=vu-\left(\int v\ⅆu\right)\\ & \text{◦}& \text{First part}\\ & & u=x\\ & \text{◦}& \text{Second part}\\ & & \mathrm{dv}=\cos \left(x\right)\\ & \text{◦}& \text{Differentiate first part}\\ & & \mathrm{du}=\frac{\ⅆ}{\ⅆx}x\\ & & \mathrm{du}=1\\ & \text{◦}& \text{Integrate second part}\\ & & v=\int \cos \left(x\right)\ⅆx\\ & & v=\sin \left(x\right)\\ & & \int \cos \left(x\right)x\ⅆx=x\sin \left(x\right)-\left(\int \sin \left(x\right)\ⅆx\right)\\ & & \text{This gives:}\\ & & -x^2\cos \left(x\right)+2x\sin \left(x\right)-2\left(\int \sin \left(x\right)\ⅆx\right)\\ \text{▫}& & \text{4. Evaluate the integral of}\sin \text{(}x\text{)}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{sin}\text{rule}\\ & & \int \sin \left(x\right)\ⅆx=-\cos \left(x\right)\\ & & \text{This gives:}\\ & & -x^2\cos \left(x\right)+2x\sin \left(x\right)+2\cos \left(x\right)\end{array}$