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I created this question, and every other question in Multiple Integration, for my dissertation Computer-Aided Assessment of Multiple Integration'.

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constant in front of function

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Evaluate the integral

\n

\$\\int_{\\var{b}}^{\\var{d}} \\int_{\\var{a}}^{\\var{c}} \\var{k} \\simplify{x^{n}} \\simplify{y^{m}} \\, \\mathrm{d}x \\, \\mathrm{d}y \\,. \$

We can evaluate this iterated integral by straight-forward integration.

\n

First integrate with respect to $x$ while treating $\\simplify{y^{m}}$ as a constant.
\\begin{align}
\\int_{\\var{b}}^{\\var{d}}\\int_{\\var{a}}^{\\var{c}}\\var{k}\\simplify{x^{n}}\\simplify{y^{m}}  \\,\\mathrm{d}x\\,\\mathrm{d}y
&=\\int_{\\var{b}}^{\\var{d}} \\, \\var{k}\\frac{1}{\\var{n+1}}x^{\\var{n+1}}\\bigg|_{x=\\var{a}}^{x=\\var{c}} \\, \\simplify{y^{m}}  \\,\\mathrm{d}y \\\\
&=\\int_{\\var{b}}^{\\var{d}}\\simplify[fractionNumbers]{{k1}}(\\var{c}^{\\var{n+1}}-\\var{a}^{\\var{n+1}})\\simplify{y^{m}}  \\,\\mathrm{d}y \\\\
&=\\int_{\\var{b}}^{\\var{d}}\\simplify[fractionNumbers]{{k1}}\\cdot \\simplify{{k2}y^{m}}  \\,\\mathrm{d}y
\\end{align}
Now we can just integrate with respect to $y$ as usual.
\\begin{align}
\\int_{\\var{b}}^{\\var{d}}\\simplify[fractionNumbers]{{k3}}\\simplify{y^{m}}  \\,\\mathrm{d}y
&= \\simplify[fractionNumbers]{{k3}}\\frac{1}{\\var{m+1}}y^{\\var{m+1}}\\bigg|_{y=\\var{b}}^{y=\\var{d}}
= \\simplify[fractionNumbers]{{k4}}(\\var{d}^{\\var{m+1}}-\\var{b}^{\\var{m+1}})
= \\simplify[fractionNumbers]{{k4}}\\times\\var{k5}
= \\simplify[fractionNumbers]{{k4*k5}}
\\end{align}

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