Using substitution to combine integrals

[sara_87]

Homework Statement

I want to combine the 2 integrals:

$$\int_{a}^{b}(x-3)f(x)dx+\int_{-b}^{-a}(x-3)f(x)dx$$

Homework Equations

given:
f(x) is an even function

The Attempt at a Solution

swap the limits in the second integral:

$$\int_{a}^{b}(x-3)f(x)dx-\int_{-a}^{-b}(x-3)f(x)dx$$

use a substitution for the second integral:
let u=-x

since f(x) is even, we have f(-u)=f(u)
so:

$$\int_{a}^{b}(x-3)f(x)dx-\int_{a}^{b}(-u-3)f(u)(-du)$$

$$\int_{a}^{b}(x-3)f(x)dx+\int_{a}^{b}(-u-3)f(u)(du)$$


now I'm stuck. can i say this:
let u=x (sounds wrong since u=-x)
so:
$$\int_{a}^{b}(x-3)f(x)dx+\int_{a}^{b}(-x-3)f(x)(dx)$$

which then can be easily combined with the first integral (now that the limits of integration are the same.

Any help will be very much appreciated. thank you

 

[sara_87]

is the latex code shown ?
I typed it in but for some reason it is not shown on my computer. here it is anyway:

Homework Statement

I want to combine the 2 integrals:

\int_{a}^{b}(x-3)f(x)dx+\int_{-b}^{-a}(x-3)f(x)dx

Homework Equations

given:
f(x) is an even function

The Attempt at a Solution

swap the limits in the second integral:

\int_{a}^{b}(x-3)f(x)dx-\int_{-a}^{-b}(x-3)f(x)dx

use a substitution for the second integral:
let u=-x

since f(x) is even, we have f(-u)=f(u)
so:

\int_{a}^{b}(x-3)f(x)dx-\int_{a}^{b}(-u-3)f(u)(-du)

\int_{a}^{b}(x-3)f(x)dx+\int_{a}^{b}(-u-3)f(u)(du)

now I'm stuck. can i say this:
let u=x (sounds wrong since u=-x)
so:
\int_{a}^{b}(x-3)f(x)dx+\int_{a}^{b}(-x-3)f(x)(dx)

which then can be easily combined with the first integral (now that the limits of integration are the same.

Any help will be very much appreciated. thank you

 

[Sybren]

You already solved it! You just need to realize that $u$ is a dummy variable, as much so as $x$ in the first integral. You can therefore replace the $u$ by whatever you like, an $x$ for example..

 

[AfterSunShine]

You almost did it.
But you should know that definite integrals are independent of the variables.

that is : $$\int_a^b f(u) du = \int_a^b f(x) dx$$

So?

 

[sara_87]

I see :)
so, i can write this as
[tex]\int_{a}^{b}(-6)f(x)dx[tex][/tex][/tex]