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

$\=x \left(-\cos \left(x\right)\right)-\int 1\cdot \left(-\cos \left(x\right)\right) \mathit{\ⅆ}x$

$\=-x \cos \left(x\right)\+\int \cos \left(x\right) \mathit{\ⅆ}x$

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

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

Figure 6.1.1(a)   Application of the Integration Methods tutor

Figure 6.1.1(b)   Integration Methods tutor after parts integration invoked

$\int x \sin \left(x\right) \mathit{\ⅆ}x$$\stackrel{\text{assign to a name}}{\to }$$q$

$\mathrm{IntegrationTools}:-\mathrm{Parts}\left(q\,x\right)$

$-x\cos \left(x\right)-\left(\∫\left(-\cos \left(x\right)\right)\ⅆx\right)$

Tools≻Tasks≻Browse: Calculus - Integral≻Methods of Integration≻Parts

Table 6.1.1(a)   Integration by Parts task template