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Antiderivatives

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rebel 

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rebelmaths

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$f(x) = \\var{a}x - \\var{b}$

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$\\int f(x) dx$  =

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$f(x) = \\frac{1}{\\var{c}}+ \\frac{2}{\\var{d}}x^2 - \\frac{3}{\\var{f}}x^3$

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$\\int f(x) dx$  =

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$f(x) = e^\\var{g}$

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$\\int f(x) dx$  =

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$f(x) = e^{\\var{g}x}$

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$\\int f(x) dx$  =

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Note that \\[\\begin{eqnarray*}&\\int& \\;a\\;dx&=&ax+C \\text{  (for any constant } a)\\\\ &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &\\int& \\;\\sin(ax)\\;dx &=& -\\frac{1}{a}\\cos(ax)+C\\\\ &\\int& \\;e^{ax}\\;dx &=& \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]

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For each of the following functions $f(x)$, find $\\int f(x) dx$ 

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Use the letter C to represent an unknown constant. 

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Note that \\[\\begin{eqnarray*}&\\int& \\;a\\;dx&=&ax+C \\text{  (for any constant } a)\\\\ &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &\\int& \\;\\sin(ax)\\;dx &=& -\\frac{1}{a}\\cos(ax)+C\\\\ &\\int& \\;e^{ax}\\;dx &=& \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]

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Don't forget to include the unknown constant C.

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(a) 

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$f(x) = \\var{a}x - \\var{b}$

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Noting that $\\var{a}x = \\var{a}x^1$ and using $\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C$ with $n=1$ we obtain

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$\\int f(x)dx = \\frac{\\var{a}}{2}x^2 - \\var{b}x+C= \\simplify{{a/2}x^2 - {b}x+C}$

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(b)

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$f(x) = \\frac{1}{\\var{c}}+ \\frac{2}{\\var{d}}x^2 - \\frac{3}{\\var{f}}x^3$

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$\\int f(x) dx  =\\frac{1}{\\var{c}}x+ \\frac{2}{\\var{d}}\\frac{x^3}{3} - \\frac{3}{\\var{f}}\\frac{x^4}{4}+C=\\simplify{x/{c}+2x^3/(3{d})-3x^4/(4{f})+C}$

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(c)

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Since $ e^\\var{g}$ is simply a constant number, we obtain $ \\int e^\\var{g}dx = e^\\var{g}x+C$

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(d)

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$f(x) = e^{\\var{g}x}$

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 Using our rule $\\int e^{ax}\\;dx = \\frac{1}{a}e^{ax}+C$ we immediately obtain

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$\\int f(x) dx =\\frac{1}{\\var{g}}e^{\\var{g}x}+C$

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