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5 indefinite integrals containing exponential functions

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rebelmaths

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Integrate $f(x)=e^{\\var{a1}x}$ with respect to $x$.

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Integrate $f(x)=\\var{a1}e^{\\var{a2}x}$ with respect to $x$.

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Integrate $f(x)=\\var{a3}\\exp(\\var{a4}x)+\\var{a1}\\exp(\\var{a5}x)$ with respect to $x$.

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Integrate $f(x)=\\dfrac{2}{\\var{a5}}\\exp(\\var{a1}x)+\\dfrac{1}{\\var{a3}}\\exp(\\var{a4}x)$ with respect to $x$.

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The basic results are

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$\\int(e^{x})dx=e^{x}+c$

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$\\int(e^{kx})dx=\\dfrac{1}{k}e^{kx}+c$

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The answers should all follow directly from the standard result $\\int(e^{kx})dx=\\dfrac{1}{k}e^{kx}+c$

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For example

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(d)

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$\\int \\dfrac{2}{\\var{a5}}e^{\\var{a1}x}+\\dfrac{1}{\\var{a3}}e^{\\var{a4}x} \\; dx=\\dfrac{2}{\\var{a5}}\\dfrac{e^{\\var{a1}x}}{\\var{a1}}+\\dfrac{1}{\\var{a3}}\\dfrac{e^{\\var{a4}x}}{\\var{a4}}+c=\\simplify{2e^({a1}x)/({a1*a5})+e^({a4}x)/({a3*a4})+c}$

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