Question

How do you evaluate the definite integral `int (2x) dx` from `[2,3]`?

Answer 1

`5`

Explanation:

in terms of the indefinite integral, use the Power Rule for integration:

`int x^n \ dx = (x^(n+1))/(n+1) + C`

And, with constant `alpha`:

`int alpha x^n \ dx = (alpha \ x^(n+1))/(n+1) + C`

Or, if you like, lift the constant outside the integration:

`int alpha x^n \ dx = alpha int x^n \ dx`

`=alpha ( \ x^(n+1))/(n+1) + C = (alpha \ x^(n+1))/(n+1) + C`

I'm labouring this, deliberately.

So

`int 2 x \ dx`

`= 2 int x^color(red)(1) \ dx`

from the Power Rule
`=2 ( x^(1+1))/(1+1) + C`

`=x^2 + C qquad triangle`

Finally, if in doubt, differentiate your result in `triangle`, because differentiation and integration are like inverse processes

`d/dx (x^2 + C) = d/dx (x^2) + d/dx(C) = 2x + 0 = 2x` Voila!!

Now for the definite integral

`int_2^3 2 x \ dx`

`= 2 int_2^3 x \ dx`

from the Power Rule
`= 2 [ x^2/2 ]_2^3`

`= [ x^2 ]_2^3`

`= 9 - 4 = 5`