Integration by change of variable

See also: Integration, Change of variable

In Mathematical, the change of variable is a process which consists in replacing a Variable or even a function by another function of this one or another parameter. This process is one of the principal tools for the resolution of Intégrale S, in analyzes.

Principle

It is the rule of integration which rises from the Théorème of derivation of the made up functions. That is to say two derivable functions $f,\; g$ and knowing, by the definition of Intégrale, that

$\backslash \; int\; f\text{'}\; (X)\; \backslash \; mathrm\; dx\; =\; \backslash \; int\; \backslash \; mathrm\; df\; (X)\; =\; F\; (X)\; +\; C$

then this theorem makes it possible to obtain

$\backslash \; int\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; mathrm\; D\}\; \{\backslash \; mathrm\; D\; G\; (X)\}\; F\; (G\; (X))\; \backslash \; cdot\; \backslash \; frac\; \{\backslash \; mathrm\; D\}\; \{\backslash \; mathrm\; dx\}\; G\; (X)\; \backslash \; right)\; \backslash \; mathrm\; dx\; =\; \backslash \; int\; \backslash \; mathrm\; D\; \backslash \; circ\; G\; (X)\; =\; F\; \backslash \; circ\; G\; (X)\; +\; C$

Example

That is to say to calculate

$\backslash \; int\; 2x\; \backslash \; cos\; (x^2)\; \backslash \; mathrm\; dx$

If one poses the change of variable $u=x^2$ and thus $\backslash \; mathrm\; du=2x\; \backslash \; mathrm\; dx$ then

$\backslash \; int\; 2x\; \backslash \; cos\; (x^2)\; \backslash \; mathrm\; dx\; =\; \backslash \; int\; \backslash \; cos\; (U)\; \backslash \; mathrm\; of\; =\; \backslash \; sin\; (U)\; +\; C\; =\; \backslash \; sin\; (x^2)\; +\; C$

Theorem

Statement

That is to say $f$ a continuous numerical function, and $\backslash \; varphi\; (T)$ a function of class $\backslash \; mathcal\; C^1$ (i.e. derivable and whose derivative is continuous) on a Intervalle $$ whose image is contained in the field of definition of $f$.

Then

$$

\ int_ {\ varphi (a)} ^ {\ varphi (b)} F (X) \, \ mathrm dx = \ int_ {has} ^ {B} F (\ varphi (T)) \ varphi' (T) \, \ mathrm dt

Demonstration

$f$ being continuous, one considers a primitive $F$ $f$ on $D$ the whole of definition of $f$. The function $F\; \backslash \; circ\; \backslash \; varphi$ is then derivable, like made up of two derivable functions and one a:

$(F\; \backslash \; circ\; \backslash \; varphi)\; \text{'}=\; (F\; \backslash \; circ\; \backslash \; varphi)\; \backslash \; times\; \backslash \; varphi\text{'}$

From where

$\backslash \; int\_\; \{has\}\; ^\; \{B\}\; F\; (\backslash \; varphi\; (T))\; \backslash \; varphi\text{'}\; (T)\; \backslash ,\; \backslash \; mathrm\; dt=\; \backslash \; int\_\; \{has\}\; ^\; \{B\}\; ((F\; \backslash \; circ\; \backslash \; varphi)\; \backslash \; times\; \backslash \; varphi\text{'})\; (T)\; \backslash ,\; \backslash \; mathrm\; dt$

$=\; \backslash \; int\_\; \{has\}\; ^\; \{B\}\; (F\; \backslash \; circ\; \backslash \; varphi)\; \text{'}(T)\; \backslash ,\; \backslash \; mathrm\; dt$

$=\; \backslash \; left\; \backslash \; varphi\; \backslash \; right\_a^b$

$=F\; (\backslash \; varphi\; (b))\; -\; F\; (\backslash \; varphi\; (a))$

$=\; \backslash \; int\_\; \{\backslash \; varphi\; (a)\}\; ^\; \{\backslash \; varphi\; (b)\}\; F\; (X)\; \backslash ,\; \backslash \; mathrm\; dx$

Traditional changes of variables

• For the functions comprising of the circular functions or hyperbolic, to see the Rules of Bioche.

• For calculer
  $\backslash \; int\; \{F\; \backslash \; left\; (X,\; \backslash \; sqrt\; \{\backslash \; frac\; \{ax+b\}\; \{cx+\; \backslash \; mathrm\; D\}\}\; \backslash \; right)\; \backslash \; mathrm\; \{\backslash \; mathrm\; D\}\; X\}$,
  where $f$ is a rational Fraction in two variables, $n$ a natural entirety and $a$, $b$, $c$ and $d$ four realities given, one pose
  $u=\; \backslash \; sqrt\; \{\backslash \; frac\; \{ax+b\}\; \{cx+\; \backslash \; mathrm\; D\}\}$:
  le change of variable will always give a rational fraction in $u$; it is then enough to break up it into simple elements to integrate.

Case of the multiple integrals

Losque $f$ is a functions of several variables, in addition to the change of the field of integration one uses the jacobien transformation “in the place” of $\backslash \; varphi\text{'}$. The jacobien is the determining of the Matrice jacobienne. One gives here the explicit formulation of the change of variable and the reader will refer to the article on the integral multiples or the Matrice jacobienne for more precise details on these concepts:

$\backslash \; iint\_D\; F\; (X,\; there)\; \backslash ;\; \backslash \; mathrm\; dx\; \backslash \; mathrm\; Dy\; =\; \backslash \; iint\_T\; F\; \backslash \; bigl\; (\backslash \; phi\; (U,\; v),\; \backslash \; Psi\; (U,\; v)\; \backslash \; bigr)\; \backslash \; left|\backslash \; frac\; \{\backslash \; share\; (\backslash \; phi,\; \backslash \; Psi)\}\{\backslash \; share\; (U,\; v)\}(U,\; v)\; \backslash \; right|~\; \backslash \; mathrm\; of\; the\; \backslash \; mathrm\; dv$.

See too

• Integral calculus

• Change of variable (algebraic simplification)
• Composition of functions

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