Question

You know from class that, for any real numbers ?,?,?a,b,c such that ?<?<?a<c<b and for any continuous function ?f, the following equalities hold:

∫???(?)??+∫???(?)??=∫???(?)??,∫acf(x)dx+∫cbf(x)dx=∫abf(x)dx,

and

∫???(?)??=−∫???(?)??.∫baf(x)dx=−∫abf(x)dx.

Use these two facts to prove that, for any continuous function ?f,

∫???(?)??+∫???(?)??=∫???(?)??,∫stf(x)dx+∫trf(x)dx=∫srf(x)dx,

for all ?,?,?∈ℝs,t,r∈R (that is, in each of the following cases: ?<?<?s<r<t, or ?<?<?t<r<s, or ?<?<?t<s<r, or ?<?<?r<s<t, or ?<?<?r<t<s).

Answer 1

Let us discuss each of the case separately:

• (s<r<t) In this case,
  
• (t<r<s) In this case,
  
• (t<s<r) In this case,
  
• (r<s<t) In this case,
  
• (r<t<s) In this case,