Find constants and antiderivative problem

[Beeorz]

Homework Statement

Find constants c1 and c2 such that F(x) = c1xsinx+c2cosx is an antiderivative of f(x) = 4xcos(x)+3sin(x)

Homework Equations

d/dx cosx = -sinx
d/dx sinx = cosx

The Attempt at a Solution

Don't know how to setup or handle a problem such as this. Prolly much simplier than I'm making it out to be, but here's my attempt:

(-4(x^2)sinx)/(2) + 3cosx
-2(x^2)sinx + 3cosx

Any advice would help. I know my method is wrong.

 

[rock.freak667]

You don;t integrate a product like that. Try using integration by parts

$$\int u \frac{dv}{dx} dx =uv- \int v \frac{du}{dx} dx$$

 

[Beeorz]

dunno if I understand that correctly...but here's what I think it means to do:

4xcosx-(cosx)(4) ??

 

[d_leet]

You want to choose c1 and c2 such that F'(x) is equal to f(x), why not differentiate F(x) set the derivative equal to f(x) and see if you can figure out what the constants should be?

 

[Beeorz]

F'(x) = c1xsinx+c2cosx

=c1sinx+c1xcosx-c2sinx

c1sinx+c1xcosx-c2sinx=c1xcosx+sinx(c1-c2)
c1-c2 = (4xcosx+3sinx-c1xcosx) / (sinx)

now what? doesn't seem like anything simplifies

 

[rock.freak667]

c1xcosx+sinx(c1-c2) = 4xcos(x)+3sin(x)

Right?

equate coefficients now.

i.e. coefficient of sin(x) in the left side = Coefficient of sin(x) on the right side.