- #1

diredragon

- 323

- 15

## Homework Statement

##f(x) =

\begin{cases}

-\frac{1}{1+x^2}, & x \in (-\infty,1) \\

x, & x \in [1,5]\setminus {3} \\

100, & x=3 \\

\log_{1/2} {(x-5)} , & x \in (5, +\infty)

\end{cases}##

For a given function determine the truth of the folowing statements and give a brief explanation:

a) Function ##F(x)=\arctan {1/x}## is one integral solution of the funtion ##f(x)## on an interval of ##(-1,1)##

b) Function f is integrable on ##[-3,5]##

c) Function f is integrable on ##[4,10]##

d) ##\int_{-1}^{1} \tan x f(x) \, dx=0##

## Homework Equations

3. The Attempt at a Solution [/B]

a) Function ##F(x)=\arctan {1/x}## is one integral solution of the funtion ##f(x)## on an interval of ##(-1,1)##

False. The solution of this integral on a provided interval is ##F(x)=-\arctan x + c##

b) Function f is integrable on ##[-3,5]##

Yes. It has 3 points of break and is continuous on intervals in between so yes. Not sure about this one so i would be grateful for your reply

c) Function f is integrable on ##[4,10]##

Yes. The same reasoning as above.

d) ##\int_{-1}^{1} \tan x f(x) \, dx=0##

Yes, the function is odd.