Why is du/dx treated as a fraction in integration by substitution?

In summary, integration by substitution involves using the notation of derivatives and differentials, which may not be true fractions, but can still be treated as such in order to solve the problem. The use of "dy/dx" as a fraction is a notational device, but it still holds true in terms of finding the derivative.
  • #1
Cheman
235
1
Integration by substitution...

Accroding to my notes, when performing integration by substitution, du/dx= f'(x), and therefore du = f'(x)*dx. But how is this possible? We are treatnig dy/dx as if it were a fraction - but in essence it is not! So why is this statement still true?

Thanks. :smile:
 
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  • #2
For better or worse, the notation used for various concepts has been chosen to appear as if you're doing arithmetic. The du and dx mean different things in the different equations.
 
  • #3
Yes, you are quite correct that dy/dx is NOT a fraction. But it IS a limit of a fraction. It can be "treated" like a fraction since we can go back "before" the limit, use the fraction properties and the take the limit again. In particular, most calculus books define the "differential" dx, basically as a notational device, and then define
dy= f'(x) dx. Strictly speaking, these "dy" and "dx" are NOT the "dy" and "dx" in "dy/dx" since that is not really a fraction but since dy= f '(x)dx (in terms of differentials), dy/dx = f '(x) and THIS dy/dx really is a fraction (although a symbolic one rather than a fraction of numbers or algebraic terms).
 

FAQ: Why is du/dx treated as a fraction in integration by substitution?

What is integration by substitution?

Integration by substitution is a technique used in calculus to simplify integrals by substituting a new variable for the original variable in the integrand. This allows for easier integration and can often lead to a closed form solution.

When should integration by substitution be used?

Integration by substitution should be used when the integrand contains a function that can be simplified by using a substitution. This is often the case when the integrand contains a composite function, such as f(g(x)), where g(x) can be substituted with u.

How do you perform integration by substitution?

The general steps for performing integration by substitution are:
1. Identify a function u that can be substituted for part of the integrand
2. Rewrite the integrand in terms of u
3. Find the derivative of u, du
4. Substitute u and du into the integral
5. Simplify and solve the integral in terms of u
6. Substitute back in the original variable.

What is the difference between u-substitution and integration by parts?

U-substitution and integration by parts are both techniques used for simplifying integrals, but they are used for different types of integrands. U-substitution is used when the integrand contains a composite function, while integration by parts is used when the integrand contains a product of two functions.

Are there any tips for choosing the right substitution?

Choosing the right substitution can often be trial and error, but there are some tips that can make it easier. Look for expressions within the integrand that resemble the derivative of a function. Also, try to choose a substitution that will eliminate a variable or simplify the integrand in some way.

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