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

Question

2488 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-09-12T19:52:19

$5$

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
$= 2 [ x^2/2 ]_2^3$

$= [ x^2 ]_2^3$

$= 9 - 4 = 5$

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