Integration: When to multiply by one or add zero?

[Batcher]

I have seen several functions be integrated by multiplying by a form of one or by adding a form of zero. When is it advantageous do do one of these things? Are there any example problems (Calc I or II) in which I can try these techniques?

 

[Mark44]

I have seen several functions be integrated by multiplying by a form of one or by adding a form of zero. When is it advantageous do do one of these things? Are there any example problems (Calc I or II) in which I can try these techniques?

Calculus and algebra textbooks have lots of examples of these techniques. A very simple example is the addition of 1/2 and 1/3.
$\frac 1 2 + \frac 1 3 = \frac 1 2 \frac 3 3 + \frac 1 3 \frac 2 2 = \frac 3 6 + \frac 2 6 = \frac 5 6$
In the second expression above, I multiplied 1 in the form of 3/3 and 2/2 to get common denominators. In more complicated problems, something similar is done so as to be able to combine fractions. You can always multiply by 1 without changing the value of the expression being multiplied.

A simple example of adding zero is in completing the square.
$y = x^2 + 4x = x^2 + 4x + (4 - 4) = x^2 + 4x + 4 - 4 = (x + 2)^2 - 4$
This example involves a function whose graph is a parabola. The second expression shows zero being added. Completing the square allows one to find the vertex of the parabola. Similar examples are done in calculus and subsequent areas of mathematics, such as Laplace transforms. You can always add zero to an expression without changing its value.

 

[mathman]

I have seen several functions be integrated by multiplying by a form of one or by adding a form of zero. When is it advantageous do do one of these things? Are there any example problems (Calc I or II) in which I can try these techniques?

Could you give some examples. Your statement is too vague.

 

[FactChecker]

@Mark44 's examples are very good. They are not really related to integration. Your question asks about integration. If you want an answer about integration then you will need to give examples.