Question

How do you integrate two variables?

Answer 1

Hint: A two-dimensional region can be integrated using double integrals. They enable us to, among other things, calculate the volume beneath a surface.

Complete step by step Solution:
Two variables are multiplied together in the integration of two variables, sometimes referred to as integration by parts.
Assuming two \[p\] and \[q\] functions of the variable \[x\] and both the algebraic function.
Then we use the formula \[\int {pqdx} = p\int {qdx} - \int {\dfrac{{dp}}{{dx}}\left( {\int {qdx} } \right)} + c \] to find the integration value.
In this formula we can take \[p\] as the first function and \[q\] as the second function.
As both the functions are functions of \[x\] then we can choose any one of the functions as the first function and the other function as the second function.
If we have both the functions are different functions like exponential, algebraic, trigonometric etc.
Then we choose the first and second functions with the ILATE rule.
ILATE rule is used for finding the first and second functions for this formula.
The full form of ILATE is inverse, logarithm, algebraic, trigonometric, and exponent.
After successfully choosing the first and second functions of \[x\] we solve the expression to get the final expression.

Note: Students get confused when in a given expression they have both the functions are the same form, like both are exponential, or both are trigonometric then they have a doubt which chooses the first and second function. We can take any one of them as the first function and the other function as the second function.