23 Integration
Consider the function \(\, F(x) = 3x^2 + 7x - 2\,\) and we want to find its derivatives as \(f(x)\), that is \[\frac {d F(x)}{dx} = f(x)\]
\[F'(x) = 6x + 7 = f(x)\]
\(F(x) = 3x^2 + 7x - 2\,\) is called an anti-derivative.
\(F(x) = 4x^3 - 7x^2 + 12x - 4\)
\(\frac {d}{dx}F(x) = f(x) = 12x^2 - 14x + 12\)
The anti-derivative of \(f(x) = 12x^2 - 14x + 12\) is
\begin {align*} & = \frac {12 x^{2+1}}{2 + 1} - \frac {14x^{1+1}}{1 + 1} + \frac {12 x^1}{1} + c\\\\ & = \frac {12}{3}x^3 - \frac {14}{2}x^2 + 12x + c\\\\ & = 4x^3 - 7x^2 + 12x + c \end {align*}
\(c\) is the constant of integration.
Definition 23.1. A function \(F(x)\) is an anti-derivative of \(f(x)\) if \(\frac {dF(x)}{dx}=f(x)\).
If \(F(x)\) is an anti-derivative of \(f(x)\) then so is \(F(x)+c\) for any constant \(c\), since differentiating a constant gives zero. Conversely, any two anti-derivatives of the same function differ only by a constant — if \(F'=G'\) then \((F-G)'=0\), and a function with zero derivative everywhere is constant.
Note 23.2. That is why the arbitrary constant is not optional decoration. \(\int f(x)\,dx\) denotes not one function but a whole family of them, every member differing from the next by a vertical shift, and \(+c\) is what says so. Omitting it gives one member of the family and claims it is all of them.
The constant disappears only in a definite integral, where it cancels in the subtraction \(F(b)-F(a)\) — as the first example below shows explicitly.
The table below can be read from right to left to get anti-derivatives.
| \(f(x)\) | \(F(x)\) an anti-derivative | |||
| 1. | \(K\) (constant) | \(Kx + c\) | ||
| 2. | \(x\) | \(\frac {x^2}{2} + c\) | ||
| 3. | \(x^2\) | \(\frac {x^3}{3} + c\) | ||
| 4. | \(x^n \) | \( \frac {x^{n + 1}}{n + 1} + c\) | ||
| 5. | \(\sin mx\) | \(\frac {-\cos mx}{m} + c\) | ||
| 6. | \(\cos mx \) | \(\frac {\sin mx}{m} + c\) | ||
| 7. | \(e^{mx}\) | \(\frac {e^{mx}}{m} + c\) | ||
| 8. | \(\frac {1}{x}\) | \(\ln \left |x\right | + c\) | ||
\[A = \lim \limits _{dx \rightarrow 0} \sum ^b_a f(x)\,dx = \int ^b_a f(x)\,dx\]
23.2 Methods of Integration
23.2.1 Integration of powers
23.2.2 Integration by Parts
23.2.3 Integration by Substitution
23.3 Trigonometric Substitution
23.4 Partial Fractions
23.5 Area Under the Curve
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