Antidifferentiation (Part 1)

1. Derivatives Properties Review. Let {F(x)} be given. We recall the derivative
(1)\displaystyle  \frac{\mathrm{d} F(x)}{\mathrm{d} x}=f(x)
is defined by
(2)\displaystyle  f(x) = F'(x) = \lim_{h\rightarrow 0}\frac{F(x+h)-F(x)}{h}.
We have several rules for differentiation:
Power Rule: If {F(x)=x^{n}}, then {F'(x)=nx^{n-1}};
Product Rule: We have {[F(x)G(x)]'=F'(x)G(x)+F(x)G'(x)};
Quotient Rule: We have {[F(x)/G(x)]'=[F'(x)G(x)-G'(x)F(x)]/G(x)^{2}};
Composition Rule: \displaystyle{\frac{\mathrm{d}}{\mathrm{d} x}F(G(x))=F'(G(x))G'(x)}.

We want to introduce an “inverse” of differentiation. We will use the notation
(3)\displaystyle  \frac{\mathrm{d} f(x)}{\mathrm{d} x}=f'(x)

2. Antidifferentiation. A function {F(x)} is an “Antiderivative” of {f(x)} on an interval {I} (usually it’s an open interval) iff
(4)\displaystyle  F'(x)=f(x).

Example 1. Let {f(x)=3x}. What is its antiderivative?

Well, we see its of the form
(5)\displaystyle  F(x)=ax^{2}
We take its derivative
(6)\displaystyle  F'(x)=2ax
Set {F'(x)} to be equal to {f(x)}
(7)\displaystyle  2ax=3x
and solve for {a}
(8)\displaystyle  a=\frac{3}{2}.
(9)\displaystyle  F(x)=\frac{3x^{2}}{2}
is the antiderivative for {f(x)}.

Note that {G(x)=F(x)+C} is also an antiderivative for {f(x)}, where {C} is any constant.

Example 2. Consider
(10)\displaystyle  g(x)=\cos(x).
What is its antiderivative?

We recall
(11)\displaystyle  \frac{\mathrm{d}}{\mathrm{d} x}\sin(x)=\cos(x)
which implies
(12)\displaystyle  G(x)=\sin(x)+C,
for some constant {C}, is the antiderivative.

Example 3. Take
(13)\displaystyle  h(x)=\frac{1}{x}+2\mathrm{e}^{2x}
and find its antiderivative.

We see that
(14)\displaystyle  \frac{\mathrm{d}}{\mathrm{d} x}\ln|x|=\frac{1}{x}.
We also remember that
(15)\displaystyle  \frac{\mathrm{d}}{\mathrm{d} x}\mathrm{e}^{2x}=2\mathrm{e}^{2x}
which means
(16)\displaystyle  \frac{\mathrm{d}}{\mathrm{d} x}\left(\ln|x|+\mathrm{e}^{2x}\right)=\frac{1}{x}+2\mathrm{e}^{2x}.
This gives us the antiderivative
(17)\displaystyle  H(x)=\ln|x|+\mathrm{e}^{2x}+C
where {C} is some constant.


About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.
This entry was posted in Antiderivative, Integral. Bookmark the permalink.

3 Responses to Antidifferentiation (Part 1)

  1. Pingback: Antidifferentiation (Part 2: Indefinite Integration) | My Math Blog

  2. Pingback: [Index] Calculus of a Single Variable | My Math Blog

  3. Pingback: [Index] Calculus in a Single Variable | My Math Blog

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