Change of variable

Answer 1

Change of variable refers to a technique used in calculus and analysis to transform an equation or integral from one set of variables to another. This technique is useful in simplifying integrals, as well as solving differential equations.

The most common type of change of variables is when a new variable u is introduced by setting u=f(x), where f is a function of x. By doing this, the integral or equation can be rewritten in terms of u, which may be easier to integrate or solve.

For example, consider the integral:

∫x^2cos(x^3)dx

Using the change of variable, u=x^3, we can rewrite the integral as:

∫(1/3)cos(u)du

which can be readily integrated.

Another type of change of variables is when a transformation is made from one coordinate system to another, such as from Cartesian to polar coordinates. This is useful for finding the area, volume, or other properties of geometric shapes.

Overall, change of variable is a powerful tool in mathematical analysis, allowing for the simplification and solution of complex equations.