Note: This concept idea hasn't been proven yet or maybe mathematically incorrect. Any contribution would be greatly appreciated.

Basic Knowledge of Integration and Functions

In mathematics, usually we take $x$ as independent variable and $y$ as dependent variable. In General, a function is written as

$y=ax^2+bx+c$

If we differentiate the above function, we get:

$\frac{dy}{dx}=2ax+b$

i.e $dy=(2ax+b)dx$

Generally, a Integral function is written as:

$f(x)=\int (2ax+b) dx$

Note: There is a wrong perception that, Integration give area under function, but actually, Definite Integration give area under a given function $f(x)$ and Indefinite Integration gives an entire family of functions with an infinite number of members.

Therefore, it is understood that:

$f(x)= \int dy=\int (2ax+b)dx$

So, After clearing the basic knowledge, we can see how to Integrate something like this:

$\int f(x,y) dx$ (where both $x$ and $y$ are variable)

Integration of two variable function $F(x,y)$

Lets take an example function:

$f'(x,y) = 3x^2+y$

So, how to find: $\int (3x^2+y)dx$

As mentioned above, it is understood that it means:

$\frac{dy}{dx}=3x^2+y$ (and we have to find its integral)

$\implies \frac{dy}{dx}-y=3x^2$ (Rearranging the Equation)

Now we have convert the Integral Question, into a Differential Equation.

So, we can solve this using, P,Q Factor method (What is Factor Method ?)

$\frac{dy}{dx}-y=3x^2$

Let $P=(-1)$ and $Q=3x^2$

Therefore:

Integral Factor $(I.F)$ = $e^{\intP dx}$

$\implies (I.F)$ = $e^{\int-1 dx}$

$\implies (I.F)$ = $e^{-x}$

Hence our result is:

$y(I.F)=\int Q (I.F) dx$

$\implies ye^{-x}=\int 3x^2e^{-x} dx$

$\implies ye^{-x} =-3(x^2e^-x +2xe^-x +2e^-x) +C$ (using By Parts)

$\implies ye^(-x)+3x^2e^{-x}+6xe^{-x}+6e^{-x}=C$ (rearranging the equation)

$\implies \ln(e^{-x}(y+3x^2+6x+6))=\ln C$ (taking log both the sides)

$\implies -x+\ln(y+3x^2+6x+6)=A$

Hence, $f(x,y)$ = $-x+\ln(y+3x^2+6x+6)=A$

Fun Assignment: Differentiate the Above function, and you will get the original $f'(x,y)$

Mathematics

Integration of two variable function

Basic Knowledge of Integration and Functions

Integration of two variable function $F(x,y)$

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Integration of two variable function

Basic Knowledge of Integration and Functions

Integration of two variable function F(x,y)F(x,y)

Comments

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Integration of two variable function $F(x,y)$