Evaluation in Maple

$\int \sin \left(7 x\right)\sin \left(6 x\right) \mathit{\ⅆ}x$ = $\frac{1}{2}\sin \left(x\right)-\frac{1}{26}\sin \left(13x\right)$

$\begin{array}{c}\int \sin \left(7x\right)\sin \left(6x\right)\ⅆx\hfill \\ & \text{=}& \frac{\int \cos \left(x\right)\ⅆx}{2}-\frac{\int \cos \left(13x\right)\ⅆx}{2}\hfill & \left[\mathrm{rewrite}\,\sin \left(7x\right)\sin \left(6x\right)=\frac{\cos \left(x\right)}{2}-\frac{\cos \left(13x\right)}{2}\right]\\ & \text{=}& \frac{\sin \left(x\right)}{2}-\frac{\int \cos \left(13x\right)\ⅆx}{2}\hfill & \left[\cos \right]\\ & \text{=}& \frac{\sin \left(x\right)}{2}-\frac{\int \cos \left(u\right)\ⅆu}{26}\hfill & \left[\mathrm{change}\,u=13x\,u\right]\\ & \text{=}& \frac{\sin \left(x\right)}{2}-\frac{\sin \left(u\right)}{26}\hfill & \left[\cos \right]\\ & \text{=}& \frac{\sin \left(x\right)}{2}-\frac{\sin \left(13x\right)}{26}\hfill & \left[\mathrm{revert}\right]\end{array}$

Table 6.2.6(a)   Evaluation of $\int \sin \left(7 x\right)\sin \left(6 x\right) \mathit{\ⅆ}x$ by the Integration Methods tutor

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$\int \sin \left(7 x\right)\sin \left(6 x\right) \mathit{\ⅆ}x$$\stackrel{\text{show solution steps}}{\to }$$\begin{array}{lll}& & \text{Integration Steps}\\ & & \int \sin \left(7x\right)\sin \left(6x\right)\ⅆx\\ \text{▫}& & \text{1. Rewrite}\\ & \text{◦}& \text{Equivalent expression}\\ & & \sin \left(7x\right)\sin \left(6x\right)=\frac{\cos \left(x\right)}{2}-\frac{\cos \left(13x\right)}{2}\\ & & \text{This gives:}\\ & & \int \left(\frac{\cos \left(x\right)}{2}-\frac{\cos \left(13x\right)}{2}\right)\ⅆx\\ \text{▫}& & \text{2. Apply the}\mathbf{\text{sum}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{sum}}\text{rule}\\ & & \int \left(f\left(x\right)+g\left(x\right)\right)\ⅆx=\int f\left(x\right)\ⅆx+\int g\left(x\right)\ⅆx\\ & & f\left(x\right)=\frac{\cos \left(x\right)}{2}\\ & & g\left(x\right)=-\frac{\cos \left(13x\right)}{2}\\ & & \text{This gives:}\\ & & \int \frac{\cos \left(x\right)}{2}\ⅆx+\int -\frac{\cos \left(13x\right)}{2}\ⅆx\\ \text{▫}& & \text{3. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\int \frac{\cos \left(x\right)}{2}\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 \frac{\cos \left(x\right)}{2}\ⅆx=\frac{\left(\int \cos \left(x\right)\ⅆx\right)}{2}\\ & & \text{We can rewrite the integral as:}\\ & & \frac{\left(\int \cos \left(x\right)\ⅆx\right)}{2}+\int -\frac{\cos \left(13x\right)}{2}\ⅆx\\ \text{▫}& & \text{4. 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:}\\ & & \frac{\sin \left(x\right)}{2}+\int -\frac{\cos \left(13x\right)}{2}\ⅆx\\ \text{▫}& & \text{5. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\int -\frac{\cos \left(13x\right)}{2}\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 -\frac{\cos \left(13x\right)}{2}\ⅆx=-\frac{\left(\int \cos \left(13x\right)\ⅆx\right)}{2}\\ & & \text{We can rewrite the integral as:}\\ & & \frac{\sin \left(x\right)}{2}-\frac{\left(\int \cos \left(13x\right)\ⅆx\right)}{2}\\ \text{▫}& & \text{6. Apply a change of variables to rewrite the integral in terms of}u\\ & \text{◦}& \text{Let}u\text{be}\\ & & u=13x\\ & \text{◦}& \text{Isolate equation for}x\\ & & x=\frac{u}{13}\\ & \text{◦}& \text{Differentiate both sides}\\ & & \mathrm{dx}=\frac{\mathrm{du}}{13}\\ & \text{◦}& \text{Substitute the values for x and dx back into the original}\\ & & \int \cos \left(13x\right)\ⅆx=\int \frac{\cos \left(u\right)}{13}\ⅆu\\ & & \text{This gives:}\\ & & \frac{\sin \left(x\right)}{2}-\frac{\left(\int \frac{\cos \left(u\right)}{13}\ⅆu\right)}{2}\\ \text{▫}& & \text{7. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\int \frac{\cos \left(u\right)}{13}\ⅆu\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant multiple}}\text{rule}\\ & & \int Cf\left(u\right)\ⅆu=C\left(\int f\left(u\right)\ⅆu\right)\\ & \text{◦}& \text{This means:}\\ & & \int \frac{\cos \left(u\right)}{13}\ⅆu=\frac{\left(\int \cos \left(u\right)\ⅆu\right)}{13}\\ & & \text{We can rewrite the integral as:}\\ & & \frac{\sin \left(x\right)}{2}-\frac{\left(\int \cos \left(u\right)\ⅆu\right)}{26}\\ \text{▫}& & \text{8. Evaluate the integral of}\cos \text{(}u\text{)}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{cos}\text{rule}\\ & & \int \cos \left(u\right)\ⅆu=\sin \left(u\right)\\ & & \text{This gives:}\\ & & \frac{\sin \left(x\right)}{2}-\frac{\sin \left(u\right)}{26}\\ \text{▫}& & \text{9. Revert change of variable}\\ & \text{◦}& \text{Variable we defined in step}6\\ & & u=13x\\ & & \text{This gives:}\\ & & \frac{\sin \left(x\right)}{2}-\frac{\sin \left(13x\right)}{26}\end{array}$

Table 6.2.6(b)   Fully detailed annotated stepwise evaluation of $\int \sin \left(7 x\right)\sin \left(6 x\right) \mathit{\ⅆ}x$