## Math2400 one year ago can someone explain this? I don't understand my instructors steps:

1. Math2400

suppose that f(1) =2. f(4)=5, f'(4)=3, and f'' is continuous. Find the value of $\int\limits_{1}^{4} xf''(x)dx$

2. Math2400

the answer is 9 but im not sure how my prof. did it. i think he tried taking the anti-deriv. first. or used some kind of property. cuz he got something like: |dw:1433615204902:dw|

3. Math2400

but idk how he got that>< can anyone explain?

4. Math2400

@Luigi0210 would u be able help?

5. Math2400

@ganeshie8 another one if you're free :) i'm studying for my final><

6. amistre64

looks like a by parts application

7. Luigi0210

^ -.-

8. amistre64

what is the derivative of a product?

9. amistre64

lets take the derivative of the product of 2 function, f and g [fg]' = ??

10. amistre64

or, to better line up with your specifics .. [gf]' = ??

11. Math2400

g'f+gf'

12. amistre64

good [gf]' = g'f + gf' now integrate both sides

13. amistre64

[gf]' integrates back in gf and we have the sum of integration on the right $gf=\int g'f~+~\int gf'$ now, we want to know how to take the integration of gf', so lets subtract to get it all by itself ... and we end up with the by parts formulation $\int gf'=gf-\int g'f$

14. amistre64

let g=x, and f' = f''

15. amistre64

the letters we named our function are immaterial, one usual notation is to use u and v $\int uv'=uv-\int vu'$

16. ybarrap

I think that you will also need $$f'(1)=0$$ as a $$\bf{given}$$ later in this process.

17. phi

Hopefully you have leaned integration by parts (which is what Amistre was deriving) https://en.wikipedia.org/wiki/Integration_by_parts You use that "rule" to get your result

18. phi

you also need to now that $\int \frac{d^2 f(x)}{dx^2} \ dx = \frac{d f(x)}{dx} = f'(x)$

19. phi

*know

20. ybarrap

Applying the rules of @amistre64 we have $$[xf']'=x'f'+xf''=f'+xf''$$ Then $$\int [xf']'dx=xf'\\ =\int \left (x'f'+xf''\right ) dx\\ =\int \left (f'+xf''\right ) dx\\$$ From which you get what you have written above. Now evaluate. You are missing an initial condition: $$f'(1)=0$$. Using this, you'll get 9 after evaluation.