Question

Integral from 0,4 (5-2x)dx, guys whats the simplest way to work this??

Answer 1

well, do you know how to calculate the antiderivative (integral) of 5 - 2x?

Answer 2

That would be x^2+c correct?

Answer 3

No

Answer 4

or rather 5-x^2|0,2?

Answer 5

I will give you an example that is similar to this:

\(\large\color{black}{ \displaystyle \int_{ 2 }^{8}\left(7-10x\right){~}dx }\)

*[1]*
An integral of a constant *b* (with respect to x) is equal to *bx*.
(For any non-zero number *b*.)

*[2]*
An integral of any \(x^n\) (with respect to x) is \(x^{n+1}/(n+1)\),
and integral of b•\(x^n\) (when you have a coefficient b in front) is \({\rm b}x^{n+1}/(n+1)\)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So you first integrate each term:

1) The integral of 7 is 7x (based on *[1]* ).
2) The integral of -10x (which is same as -10x\(^1\),
is \({\rm -10}x^{1+1}/(1+1)=-10x^2/2=-5x^2\)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So you get:

\(\large\color{black}{ \displaystyle \int_{ 2 }^{8}\left(7-10x\right){~}dx =7x-5x^2 }\)

and then evaluate that from x=2 to x=8 (those are the limits of integration)
In other words:

\(\large\color{black}{ \displaystyle \int_{ 2 }^{8}\left(7-10x\right){~}dx =\left[7(8)-5(8)^2\right] -\left[7(2)-5(2)^2\right]=-258 }\)

Answer 6

This is an example, read through it and ask if you have any questions.

Answer 7

So what happens the the dx?

Answer 8

So the \(\displaystyle \int_{a}^{b}\) and \(\rm dx\) are just indicating that you are integrating with respect to x, from x=a to x=b.