Integral calculus

The integral calculus is the second of the ideas of the Infinitesimal calculus .

Primitives

That is to say $f\; \backslash ,$ a function defined on an interval $I\; \backslash ,$. A function $F\; \backslash ,$ is a primitive of $f\; \backslash ,$ on the interval $I\; \backslash ,$ if $F\; \backslash ,$ is derivable on $I\; \backslash ,$ and so for all $x\; \backslash ,$ of $I\; \backslash ,$, $F\text{'}\; (X)\; =\; F\; (X)\; \backslash ,$. If $f\; \backslash ,$ is a continuous function on an interval $I\; \backslash ,$, then it exists at least a function $F\; \backslash ,$ derivable on $I\; \backslash ,$ such as $f\; \backslash ,$ is the derivative of $F\; \backslash ,$ on $I\; \backslash ,$. $F\; \backslash ,$ is then a primitive of $f\; \backslash ,$ on $I\; \backslash ,$.

For example, if $f\; \backslash ,$ is defined on $\backslash \; R\; \backslash ,$ by $f\; \backslash \; colonist\; X\; \backslash \; mapsto\; 6x$, then the function $F\; \backslash ,$ definite on $\backslash \; R\; \backslash ,$ by $F\; \backslash \; colonist\; X\; \backslash \; mapsto\; 3x^2\; \backslash ,$ admits for derived $f\; \backslash ,$, and thus $F\; \backslash ,$ is a primitive of $f\; \backslash ,$ on $\backslash \; R\; \backslash ,$.

If $F\; \backslash ,$ is a primitive of $f\; \backslash ,$ on $I\; \backslash ,$, then for any constant $k\; \backslash ,$, the function $G\; \backslash ,$ definite on $\backslash \; R$ by $G\; \backslash \; colonist\; X\; \backslash \; mapsto\; F\; (X)\; +\; K\; \backslash ,$ is also a primitive of $f\; \backslash ,$ on $I\; \backslash ,$ because the derivative of a constant application is the null function. One deduces from it that if $f\; \backslash ,$ admits a primitive on $I\; \backslash ,$ then it admits an infinity of it.

Together primitives of a function on an interval

Two primitives different from the same function $f\; \backslash ,$ differ only from one constant. Indeed if $F\; \backslash ,$ and $G\; \backslash ,$ are two primitives of $f\; \backslash ,$ then $F\text{'}\; =\; G\text{'}\; =\; F\; \backslash ,$ thus $(F\; -\; G)\; \text{'}=\; 0\; \backslash ,$. $I\; \backslash ,$ being an interval, we deduce from it that there exists $C\; \backslash ,$ a constant defined on such as $F\; -\; G\; =\; C\; \backslash ,$ is $F\; =\; G\; +\; C\; \backslash ,$

That is to say $f\; \backslash ,$ a function defined on an interval $I\; \backslash ,$. If $f\; \backslash ,$ admits a primitive $F\; \backslash ,$ on $I\; \backslash ,$, then the whole of the primitives of $f\; \backslash ,$ on $I\; \backslash ,$ is the whole of the functions $G\; \backslash ,$ of the form:

$G:\; I\; \backslash \; rightarrow\; \backslash \; R\; \backslash ,$

$x\; \backslash \; mapsto\; F\; (X)\; +\; K\; \backslash ,$

where $k\; \backslash ,$ is a real constant. It is noticed that the primitives of the null function are the constant functions.

That is to say $I\; \backslash ,$ an interval, $a\; \backslash ,$ a reality of $I\; \backslash ,$ and $b\; \backslash ,$ an unspecified reality. There exists one and only one primitive $F\; \backslash ,$, of a function $f\; \backslash ,$ continues on $I\; \backslash ,$, such as $F\; (a)=b\; \backslash ,$. $F\; \backslash ,$ is called the primitive of $f\; \backslash ,$ on $I\; \backslash ,$ checking the initial condition: $F\; (a)\; =\; B\; \backslash ,$.

For example to find the primitive of $f\; (X)\; =\; 7x\; -\; 3\; \backslash ,$ checking the initial condition $F\; (1)\; =1\; \backslash ,$.

One calculates initially the general form of the primitive $F\; (X)\; =\; \{7\; \backslash \; over\; 2\}\; x^2-3x+k\; \backslash ,$.

Then one solves the equation $\{7\; \backslash \; over\; 2\}\; *1^2-3*1+k=1\; \backslash ,$ and one obtains $k=\; \{1\; \backslash \; over\; 2\}\; \backslash ,$ and thus the required primitive is $F\; (X)\; =\; \{7\; \backslash \; over\; 2\}\; x^2-3x+\; \{1\; \backslash \; over\; 2\}\; \backslash ,$.

Integral

Definition of the integral starting from the concept of primitive

That is to say $f\; \backslash ,$ a function defined on an interval $I\; \backslash ,$ and admitting primitives on $I\; \backslash ,$. Are $a\; \backslash ,$ and $b\; \backslash ,$ in $I\; \backslash ,$. That is to say $F\; \backslash ,$ a primitive of $f\; \backslash ,$ on $I\; \backslash ,$. We call integral of $a\; \backslash ,$ with $b\; \backslash ,$ of $f\; \backslash ,$, the number:

$F\; (b)\; -\; F\; (a)\; \backslash ,$

who does not depend on the choice of the primitive of $f\; \backslash ,$, since the primitives of $f\; \backslash ,$ on the interval $I\; \backslash ,$ differ from a constant function. We note this number:

$\backslash \; int\_a^b\; F\; (T)\; dt\; \backslash ,$

who is read “integral $a\; \backslash ,$ with $b\; \backslash ,$ of $f\; \backslash ,$”, and we can also note it

$\backslash \; left\_a^b\; \backslash ,$

who is read “$F\; \backslash ,$. taken between $a\; \backslash ,$. and $b\; \backslash ,$. ”

In the notation with the symbol? , $t\; \backslash ,$ play the part of a dummy variable, and we have

$\backslash \; int\_a^b\; F\; (T)\; dt=\; \backslash \; int\_a^b\; F\; (X)\; dx=\; \backslash \; ldots\; \backslash ,$,

moreover the number represented by this integral does not depend on $t\; \backslash ,$.

Let us notice if $f\; \backslash ,$ is continuous on $I\; \backslash ,$, that the application $G\; \backslash ,$ definite on $I\; \backslash ,$:

$G:\; X\; \backslash \; mapsto\; \backslash \; int\_a^x\; F\; (T)\; dt=F\; (X)\; -\; F\; (a)\; \backslash ,$

is not other than the primitive of $f\; \backslash ,$ which is cancelled in $a\; \backslash ,$ and this function $G\; \backslash ,$ is thus the only derivable function on $I\; \backslash ,$ such $G\text{'}=f\; \backslash ,$ and $G\; (a)\; =\; 0\; \backslash ,$.

We thus have

$\backslash \; int\_a^a\; F\; (T)\; dt=F\; (A)\; -\; F\; (a)=0\; \backslash ,$

Properties of the integral

Linearity of the integral

If $f\; \backslash ,$ and $g\; \backslash ,$ are two functions defined on an interval $I\; \backslash ,$ and admitting primitives on $I\; \backslash ,$, then the function $f+g\; \backslash ,$ admits also primitives on $I\; \backslash ,$ and for all $a\; \backslash ,$ and all $b\; \backslash ,$ of $I\; \backslash ,$, one a:

$\backslash \; int\_a^b\; (F\; (T)\; +g\; (T))\; dt=\; \backslash \; int\_a^b\; F\; (T)\; dt\; +\; \backslash \; int\_a^b\; G\; (T)\; dt\; \backslash ,$

Moreover, if $\backslash \; lambda\; \backslash ,$ is an unspecified reality then the function $\backslash \; lambda\; \backslash ,\; F\; \backslash ,$ admits primitives on $I\; \backslash ,$ and:

$\backslash \; int\_a^b\; \backslash \; lambda\; F\; (T)\; dt=\; \backslash \; lambda\; \backslash \; int\_a^b\; F\; (T)\; dt\; \backslash ,$

Relation of Chasles

Are $a\; \backslash ,$ and $b\; \backslash ,$ two realities of the interval $I\; \backslash ,$. If $f\; \backslash ,$ a function defined on $I\; \backslash ,$ and admitting primitives on $I\; \backslash ,$, then for all $a\; \backslash ,$, $b\; \backslash ,$ and $c\; \backslash ,$ in $I\; \backslash ,$

$\backslash \; int\_a^c\; F\; (X)\; dx=\; \backslash \; int\_a^b\; F\; (X)\; dx+\; \backslash \; int\_b^c\; F\; (X)\; dx\; \backslash ,$ (relation of Chasles)

Indeed if $F\; \backslash ,$ is a primitive of $f\; \backslash ,$ on $I\; \backslash ,$ then:

$F\; (b)\; -\; F\; (a)\; =\; (F\; (c)\; -\; F\; (a))\; +\; (F\; (b)\; -\; F\; (c))\; \backslash ,$.

By taking $a\; =\; B\; \backslash ,$ in the relation of Chasles, we obtain:

$\backslash \; int\_a^c\; F\; (X)\; dx=-\; \backslash \; int\_c^a\; F\; (X)\; dx\; \backslash ,$

indeed

$0=\; \backslash \; int\_a^a\; F\; (X)\; dx=\; \backslash \; int\_a^c\; F\; (X)\; dx+\; \backslash \; int\_c^a\; F\; (X)\; dx\; \backslash ,$

Positivity of the integral

That is to say $f\; \backslash ,$ a function defined on the interval $I\; \backslash ,$ which admits primitives on $I\; \backslash ,$, and if $a\; \backslash ,$ and $b\; \backslash ,$ is two realities in $I\; \backslash ,$ such as .

So for any reality $x\; \backslash ,$ of $\backslash \; left\; B\; \backslash \; right\; \backslash ,$, $f\; (X)\; \backslash \; geq\; 0\; \backslash ,$ then

$\backslash \; int\_a^b\; F\; (X)\; dx\; \backslash \; geq\; 0\; \backslash ,$

Indeed under this condition, any primitive of $f\; \backslash ,$ on the interval $I\; \backslash ,$ is increasing.

Consequences:

Growth of the integral

If $f\; \backslash ,$ and $g\; \backslash ,$ admit primitives on $I\; \backslash ,$ and so for all $x\; \backslash ,$ in $\backslash \; left\; B\; \backslash \; right\; \backslash ,$, $f\; (X)\; \backslash \; Leq\; G\; (X)\; \backslash ,$ then

$\backslash \; int\_a^b\; F\; (X)\; dx\; \backslash \; Leq\; \backslash \; int\_a^b\; G\; (X)\; dx\; \backslash ,$

(it is enough to pose $h=g\; \backslash \; -\; F\; \backslash ,$ and to use the positivity and the linearity of the integral)

Inequality of the average

If there exists $m\; \backslash ,$ and $M\; \backslash ,$ of realities such as for all $x\; \backslash ,$ in $\backslash \; left\; B\; \backslash \; right\; \backslash ,$, $m\; \backslash \; Leq\; F\; (X)\; \backslash \; Leq\; M\; \backslash ,$, then

$m\; (Ba)\; \backslash \; Leq\; \backslash \; int\_a^b\; F\; (X)\; dx\; \backslash \; Leq\; M\; (Ba)\; \backslash ,$

If there exists a reality $M\; \backslash ,$ such as for all $x\; \backslash ,$ in $\backslash \; left\; B\; \backslash \; right\; \backslash ,$, $\backslash \; left|F\; (X)\; \backslash \; right|\; \backslash \; Leq\; M\; \backslash ,$, then

$\backslash \; left|\backslash \; int\_a^b\; F\; (X)\; dx\; \backslash \; right|\backslash \; Leq\; M\; (B\; -\; a)\; \backslash ,$

If there exists a reality $M\; \backslash ,$ such as for all $x\; \backslash ,$ in $I\; \backslash ,$, $\backslash \; left\; |F\; (X)\; \backslash \; right|\; \backslash \; Leq\; M\; \backslash ,$, then for all $a\; \backslash ,$ and all $b\; \backslash ,$ in $I\; \backslash ,$,

$\backslash \; left|\backslash \; int\_a^b\; F\; (X)\; dx\; \backslash \; right|\backslash \; Leq\; M|B\; -\; has|\backslash ,$

simple Form of the first theorem of the average

If $f\; \backslash ,$ is continuous on $I\; \backslash ,$, then for all $a\; \backslash ,$ and all $b\; \backslash ,$ in $I\; \backslash ,$, it exists a reality $c\; \backslash ,$ ranging between $a\; \backslash ,$ and $b\; \backslash ,$ such as:

$\backslash \; int\_a^b\; F\; (X)\; dx=f\; (c)\; (B\; -\; a)\; \backslash ,$

Median value of a function

If $f\; \backslash ,$ admits primitives on an interval $I\; \backslash ,$, if $a\; \backslash ,$ and $b\; \backslash ,$ are in $I\; \backslash ,$ such as $a\; \backslash ,$<$b\; \backslash ,$, we call median value $f\; \backslash ,$ on $\backslash \; left\; B\; \backslash \; right\; \backslash ,$, the number:

$\backslash \; frac\; \{1\}\; \{Ba\}\; \backslash \; int\_a^b\; F\; (X)\; dx\; \backslash ,$

Parity

That is to say $f\; \backslash ,$ a function which admits primitives on an interval $I\; \backslash ,$ centered into 0. If $a\; \backslash ,$ is a reality, such as $a\; \backslash ,$ and $-a\; \backslash ,$ belong to $I\; \backslash ,$, then:

• if $f\; \backslash ,$ are even, $\backslash \; int\_\; \{-\; has\}\; ^\; \{has\}\; F\; (X)\; dx=\; 2\; \backslash \; int\_0^\; \{has\}\; F\; (X)\; dx\; \backslash ,$
• if $f\; \backslash ,$ is odd, $\backslash \; int\_\; \{-\; has\}\; ^\; \{has\}\; F\; (X)\; dx=0\; \backslash ,$

Integral and surface

A particular case:

Are $a\; \backslash ,$ and $b\; \backslash ,$ two realities such as . Either $f\; \backslash ,$ a constant function on $\backslash \; left\; B\; \backslash \; right\; \backslash ,$ and or $c\; \backslash ,$ such as

for any reality $x\; \backslash ,$ of $\backslash \; left\; B\; \backslash \; right\; \backslash ,$, $f\; (X)\; \backslash ,$=$c\; \backslash ,$

Then the integral of $a\; \backslash ,$ with $b\; \backslash ,$ of $f\; \backslash ,$ is equal to $c\; (B\; \backslash ,$-$a)\; \backslash ,$ and represents the algebraic surface of the rectangle of tops $(has,\; 0)\; \backslash ,$, $(B,\; 0)\; \backslash ,$, $(B,\; c)\; \backslash ,$ and $(has,\; c)\; \backslash ,$.

Theorem:

Are $a\; \backslash ,$ and $b\; \backslash ,$ two realities such as . Either $f\; \backslash ,$ a function continues on $\backslash \; left\; B\; \backslash \; right\; \backslash ,$. That is to say $(x\_0\; \backslash ,$, $x\_1\; \backslash ,$,…, $x\_n)\; \backslash ,$ a strictly increasing succession of points sharing the segment $\backslash \; left\; B\; \backslash \; right\; \backslash ,$ in $n\; \backslash ,$ intervals length

$\backslash \; frac\; \{Ba\}\; \{N\}\; \backslash ,$

We then have for all $i\; \backslash ,$ ranging between $0\; \backslash ,$ and $n\; \backslash ,$,

$x\_i=a+i.\; \backslash \; frac\; \{Ba\}\; \{N\}\; \backslash ,$

Then the sum

$\backslash \; frac\; \{Ba\}\; \{N\}\; \backslash \; sum\_\; \{i=0\}\; ^\; \{n-1\}\; F\; \backslash \; left\; (a+i\; \backslash \; frac\; \{Ba\}\; \{N\}\; \backslash \; right)\; \backslash ,$

tends towards $\backslash \; int\_a^b\; F\; (T)\; dt\; \backslash ,$ when $n\; \backslash ,$ tends towards $+\; \backslash \; infty\; \backslash ,$.

Graphic interpretation:

This sum (called sum of Riemann) graphically represents the algebraic sum of the surfaces of the rectangles of left and is an approximate value of $\backslash \; int\_a^b\; F\; (T)\; dt\; \backslash ,$.

If $f\; \backslash ,$ is a continuous positive function on $\backslash \; left\; B\; \backslash \; right\; \backslash ,$ and if $\backslash \; mathcal\; C\; \backslash ,$ is the curve representative of $f\; \backslash ,$ in the plan brought back to an orthogonal reference mark $(O\; \backslash ,$; $i\; \backslash ,$, $j)\; \backslash ,$, $\backslash \; int\_a^b\; F\; (T)\; dt\; \backslash ,$ is the measurement of the surface of the plan delimited by $\backslash \; mathcal\; C\; \backslash ,$, the x-axis $O\; \backslash ,$ $x\; \backslash ,$ and the lines of equations $x\; \backslash ,$=$a\; \backslash ,$ and $x\; \backslash ,$=$b\; \backslash ,$. The unit of surface being the surface of the rectangle $O\; \backslash ,$ $I\; \backslash ,$ $K\; \backslash ,$ $J\; \backslash ,$.

Methods of calculating of an integral

Direct calculation using the usual primitives

Integration by parts

Theorem:

That is to say $I\; \backslash ,$ an interval. Are $f\; \backslash ,$ and $g\; \backslash ,$ two functions derivable on $I\; \backslash ,$ such as the functions $f\text{'}\; \backslash ,$ $g\; \backslash ,$ and $f\; \backslash ,$ $g\text{'}\; \backslash ,$ are continuous on $I\; \backslash ,$. That is to say $a\; \backslash ,$ a reality in $I\; \backslash ,$. Then, for any reality $x\; \backslash ,$ in $I\; \backslash ,$

$\backslash \; int\_a^x\; f^\; \{\backslash \; premium\}\; \backslash \; left\; (T\; \backslash \; right)\; G\; \backslash \; left\; (T\; \backslash \; right)\; dt=\; \backslash \; left\; T\; \backslash \; right)\; G\; \backslash \; left\; (T\; \backslash \; right)\; \backslash \; right\_a^x\; -\; \backslash \; int\_a^x\; F\; \backslash \; left\; (T\; \backslash \; right)\; g^\; \{\backslash \; premium\}\; \backslash \; left\; (T\; \backslash \; right)\; dt\; \backslash ,$

In particular:

Theorem:

Are $a\; \backslash ,$ and $b\; \backslash ,$ two realities such as . Are $f\; \backslash ,$ and $g\; \backslash ,$ two functions derivable on $\backslash \; left\; B\; \backslash \; right\; \backslash ,$ and such as the functions $f\text{'}\; \backslash ,$, $g\; \backslash ,$, $f\; \backslash ,$ and $g\text{'}\; \backslash ,$ are continuous on $\backslash \; left\; B\; \backslash \; right\; \backslash ,$. Then:

$\backslash \; int\; \_\; \{has\}\; ^\; \{B\}\; f^\; \{\backslash \; premium\}\; \backslash \; left\; (T\; \backslash \; right)\; G\; \backslash \; left\; (T\; \backslash \; right)\; dt=\; \backslash \; left\; F\; \backslash \; left\; (T\; \backslash \; right)\; G\; \backslash \; left\; (T\; \backslash \; right)\; \backslash \; right\; \_\; \{has\}\; ^\; \{B\}\; -\; \backslash \; int\; \_\; \{has\}\; ^\; \{B\}\; F\; \backslash \; left\; (T\; \backslash \; right)\; g^\; \{\backslash \; premium\}\; \backslash \; left\; (T\; \backslash \; right)\; dt\; \backslash ,$

One can generalize this formula with the functions of class $C^\; \{k+1\}$

$\backslash \; int\_\; \{has\}\; ^\; \{B\}\; f^\; \{k+1\}\; (X)\; G\; (X)\; \backslash ,\; dx\; =\; \backslash \; left\; \backslash \; sum\_\; \{n=0\}\; ^\; \{K\}\; (-\; 1)\; ^\; \{N\}\; f^\; \{kN\}\; (X)\; g^\; \{N\}\; (X)\; \backslash \; right\_\; \{has\}\; ^\; \{B\}\; +\; (-\; 1)\; ^\; \{k+1\}\; \backslash \; int\_\; \{has\}\; ^\; \{B\}\; F\; (X)\; g^\; \{k+1\}\; (X)\; \backslash ,\; dx$

Integration by the method of the residues

See also: Remainder theorem

Approximate numerical calculation of an integral

One considers here the case of a function $f\; \backslash ,$ definite on $\backslash \; left\; B\; \backslash \; right\; \backslash ,$. One defines the “step” of approximation $h\; \backslash ,$ in the following way: $h\; =\; \backslash \; frac\; \{Ba\}\; \{N\}\; \backslash ,$; where $n\; \backslash ,$ determines the precision of the approximation. One defines also $x\_i\; =\; has\; +ih\; \backslash ,$.

Method of the rectangles

The method of the rectangles returns to an approximation of $f\; \backslash ,$ by a function in staircase, with $n\; \backslash ,$ “steps” length $h\; \backslash ,$. The approximate value $R\; \backslash ,$ of the integral is worth then:

$R\; =\; H\; \backslash \; sum\_\; \{I\; =\; 0\}\; ^\; \{N\; -\; 1\}\; F\; (x\_i)\; \backslash ,$.

Method of the trapezoids

See also: Method of the trapezoids

One uses a function continues closely connected per pieces approaching the function to be integrated and equal to this one on the points of the subdivision in $N\; \backslash ,$ equal subintervals of the interval of integration $\backslash \; left\; has,\; B\; \backslash \; right\; \backslash ,$ to obtain an approximation of the value of his integral on $\backslash \; left\; has,\; B\; \backslash \; right\; \backslash ,$.

By replacing by trapezoids the rectangles used previously, one obtains:

$R\; =\; H\; \backslash \; left\; \backslash \; frac\; \{F\; (a)+f\; (b)\}\; \{2\}\; +\; \backslash \; sum\_\; \{i=1\}\; ^\; \{n-1\}\; F\; (x\_i)\; \backslash \; right\; \backslash ,$.

One can determine the precision of this approximation by using the following formula:

$\backslash \; left|\; I\; -\; R\; \backslash \; right|\; \backslash \; Leq\; \backslash \; frac\; \{M\; (B\; -\; a)^3\}\; \{12n^2\}\; \backslash ,$ where $M\; \backslash ,$ is the upper limit of the absolute value of derived from order 2 of $f\; \backslash ,$ on $\backslash ,$ and $I\; \backslash ,$ the exact value of the integral.

Method of Simpson

See also: Method of Simpson

One now uses parabolas which one makes pass by three consecutive points of cutting in $2n\; \backslash ,$ segments of the interval of integration of $f\; \backslash ,$.

One is based on the following exact result where $P\; \backslash ,$ is a polynomial function of degree two:

If $a\; \backslash ,$, $b\; \backslash ,$ and $c\; \backslash ,$ are three realities such as $c\; =\; \backslash \; frac\; \{(b+a)\}\; \{2\}\; \backslash ,$, then $\backslash \; int\_\; \{has\}\; ^\; \{B\}\; P\; (X)\; \backslash ,\; dx\; =\; \backslash \; frac\; \{(Ba)\}\{6\}\; \backslash \; begin\; \{pmatrix\}\; F\; (a)+4f\; (c)+f\; (b)\; \backslash \; end\; \{pmatrix\}\; \backslash ,$ One then obtains an approximate value of $I\; \backslash ,$ with the following formula:

$R\; =\; \backslash \; frac\; \{H\}\; \{6\}\; \backslash \; left\; F\; (a)\; +\; F\; (b)\; +\; 4\; \backslash \; sum\_\; \{i=0\}\; ^\; \{n-1\}\; F\; (x\_\; \{2i+1\})\; +\; 2\; \backslash \; sum\_\; \{i=1\}\; ^\; \{n-1\}\; F\; (x\_\; \{2i\})\; \backslash \; right\; \backslash ,$ where $H\; =\; \backslash \; frac\; \{Ba\}\; \{N\}$ and $x\_k\; =\; has\; +\; K\; \backslash \; frac\; h2$

One can also determine the precision of the method here, with the following formula:

$\backslash \; left|\; I\; -\; R\; \backslash \; right|\; \backslash \; Leq\; \backslash \; frac\; \{M\; (B\; -\; a)^5\}\; \{2880n^4\}\; \backslash ,$ where $M\; \backslash ,$ is the upper limit of the absolute value of derived from order 4 of $f\; \backslash ,$ on $\backslash ,$ and $I\; \backslash ,$ the exact value of the integral.

Method of Gauss-Legendre

See also: Methods of squaring of Gauss

One also uses in numerical analysis a method based on orthogonality of the Polynômes of Legendre. for the scalar product $\backslash \; left\; \backslash \; langle\; F|G\; \backslash \; right\; \backslash \; rangle\; =\; \backslash \; int\_\; \{-\; 1\}\; ^\; \{+1\}\; F\; (X)\; G\; (X)\; \backslash ,\; dx\; \backslash ,$

It is called method of Gauss-Legendre, and makes it possible to calculate with a high degree of accuracy the integrals of sufficiently regular functions on a segment $\backslash \; left\; has,\; B\; \backslash \; right\; \backslash ,$

It is enough to carry out an application closely connected of $\backslash \; left\; has,\; B\; \backslash \; right\; \backslash ,$ on $\backslash \; left\; -1,\; +1\; \backslash \; right\; \backslash ,$, and to notice that

$\backslash \; int\_\; \{has\}\; ^\; \{B\}\; F\; (X)\; \backslash ,\; dx\; =\; \backslash \; frac\; \{(Ba)\}\{2\}\; \backslash \; int\_\; \{-\; 1\}\; ^\; \{+1\}\; F\; \backslash \; begin\; \{pmatrix\}\; \backslash \; frac\; \{b+a\}\; \{2\}\; +\; \backslash \; frac\; \{Ba\}\; \{2\}\; X\; \backslash \; end\; \{pmatrix\}\; \backslash ,\; dx\; \backslash \; approx\; \backslash \; frac\; \{B\; -\; has\}\; \{2\}\; \backslash \; sum\_\; \{K\; =\; 1\}\; ^\; \{N\}\; \{m\; \backslash \; left\; (\{x\_k\}\; \backslash \; right)\}\; F\; (\backslash \; frac\; \{B\; -\; has\}\; \{2\}\; x\_k\; +\; \backslash \; frac\; \{B\; +\; has\}\; \{2\})\; \backslash ,$

where $x\_k\; \backslash ,$ is the roots of the polynomial of Legendre of degree $n\; \backslash ,$

and where $m\; \backslash \; left\; (\{x\_k\}\; \backslash \; right)\; \backslash ,$ is the weights of these roots, which are such as the equality

$\backslash \; int\_\; \{-\; 1\}\; ^\; \{+1\}\; F\; \backslash \; begin\; \{pmatrix\}\; X\; \backslash \; end\; \{pmatrix\}\; \backslash ,\; dx\; =\; \backslash \; sum\_\; \{K\; =\; 1\}\; ^\; \{N\}\; \{m\; \backslash \; left\; (\{x\_k\}\; \backslash \; right)\; F\; (x\_k)\}\backslash ,$ is ensured for any polynomial function of degree lower or equal to $2n-1\; \backslash ,$

The first polynomials are

$P\_0\; (X)\; =1\; \backslash ,$

$P\_1\; (X)\; =x\; \backslash ,$

$P\_2\; (X)\; =1-3x^2\; \backslash ,$

…

An excellent precision is guaranteed as soon as $N\; \backslash \; Ge\; 3\; \backslash ,$. Tables make it possible to obtain the values of the points and their weights.

Example

See too

• Integral
• Table of integrals
• Integral calculus
• numerical Calculation of a Primitive integral
• Table of primitives
• Rules of Bioche

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