16 Riemann integral[19K]

All definitions and theorems needed to solve the following exercises may be found in Chap. 1 in [ 4 ] , or Chap. 6 of [ 22 ] .

E373

[19M] Let \(p\) be a polynomial (with complex coefficients); fix \(𝜃∈ℂ, 𝜃≠ 0\). Define \(f(x)=-∫_ 0^ x e^{-𝜃 t} p(t)\, {\mathbb {d}}t\). Show that \(f(x)=e^{-𝜃 x}q(x)-q(0)\) where \(q\) is a polynomial that has the same degree as \(p\). Determine the linear map (i.e. the matrix) that transforms the coefficients of \(p\) into the coefficients of \(q\); and its inverse.

Hidden solution: [UNACCESSIBLE UUID ’19N’][UNACCESSIBLE UUID ’19P’]

E373

[19Q] Note:Similar to point 8 from exercise 5.Suppose \(f_ n:[a,b]→ℝ\) are Riemann-integrable, and \(f:[a,b]→ℝ\) a generic function.

Find an example where \(f_ n→_ n f\) pointwise, \(f\) is bounded, but \(f\) is not Riemann integrable.

Show that, if the convergence is uniform, then \(f\) is Riemann integrable and

\[ \lim _{n→∞} ∫_ a^ b f_ n\, {\mathbb {d}}x= ∫_ a^ b f\, {\mathbb {d}}x\quad . \]

Hidden solution: [UNACCESSIBLE UUID ’19R’]

E373

[19S] Prerequisites:2, 1.

Let \(I⊂ ℝ\) be an interval with extremes \(a,b\). Let \(f,f_ n:I→ℝ\) be continuous non-negative functions such that \(f_ n(x)↗_ n f\) pointwise (i.e. for every \(x\) and \(n\) we have \(0≤ f_ n(x) ≤ f_{n+1}(x)\) and \(\lim _ n f_ n(x) =f(x)\)). Prove that

\[ \lim _{n→∞} ∫_ a^ b f_ n(x)\, {\mathbb {d}}x=∫_ a^ b f(x)\, {\mathbb {d}}x~ ~ . \]

(Note if the interval is open or semiopen or unbounded then the Riemann integrals are understood in a generalized sense; in this case the right term can also be \(+∞\)). Hidden solution: [UNACCESSIBLE UUID ’19T’]

The previous result is called Monotonic Convergence Theorem and holds in very general hypotheses; in the case of Riemann integrals, however, it can be seen as a consequence of the results 2 and 1.

E373

[19V]Suppose that \(f:[a,b]→ℝ\) is Riemann integrable and \(g:ℝ→ℝ\) is continuous, prove that \(g◦ f\) is Riemann integrable.

Hidden solution: [UNACCESSIBLE UUID ’19W’] [UNACCESSIBLE UUID ’19X’]

[19Y]Say which of these functions \(f:[0,1]→ℝ\) are Riemann integrable:

  1. the characteristic function of the Cantor set.

  2. \(f(0)=0\), \(f(x)=\sin (1/x)\)

  3. \(f(0)=0\) and

    \[ f(x)=\frac{1-\cos (x)}{x^ 2+|\sin (1/x)|} \]

  4. \(f(x)=0\) if \(x\) is irrational, \(f(x)=\cos (1/m)\) if \(x=n/m\) with \(n,m\) coprime. [UNACCESSIBLE UUID ’19Z’]

\(f(x)=0\) if \(x\) is irrational, \(f(x)=\sin (1/m)\) if \(x=n/m\) with \(n,m\) coprime.

[1B0]Prerequisites:Fundamental theorem of integral calculus.

Suppose that \(f:[a,b]→ℝ\) is continuous and \(g:ℝ→ℝ\) has class \(C^ 1\): prove that

\[ ∫_ a^ b f(g(t)) g'(t) \, {\mathbb {d}}t = ∫_{g(a)}^{g(b)} f(s) \, {\mathbb {d}}s \quad . \]

Hidden solution: [UNACCESSIBLE UUID ’1B2’] Note that for this result it is not necessary to assume that \(g\) is monotonic. [1B3] Prerequisites:regulated functions Sec. 13.2.

Show that a regulated function \(f:[a,b]→ℝ\) is Riemann integrable. [1B4]Prerequisites:Regulated functions Sec. 13.2.

Find a Riemann integrable function \(f:[0,1]→ℝ\) that is not regulated.

Hidden solution: [UNACCESSIBLE UUID ’1B5’] [1B6]Difficulty:*.Can there be a Riemann integrable function \(f:[0,1]→ℝ\) that is not regulated (i.e., it does not allow right and left limits) at any point? Hidden solution: [UNACCESSIBLE UUID ’1B7’] [1B8]If \(f,g:[A,B]→ℝ\) are Riemann integrable, then \(h(x)=\max \{ f(x),g(x)\} \) is Riemann integrable. [1B9]Find a lower semicontinuous function \(f:[0,1]→ℝ\), bounded, but not Riemann integrable.

[UNACCESSIBLE UUID ’1BB’] [1BC]We define the Beta function as

\[ B(x,y) = ∫_ 0^ 1 t^{x-1}(1-t)^{y-1}~ dt~ . \]

  1. Show that the integral exists (finite) if and only if \(x,y{\gt}0\).

  2. Note that \(B(x,y)=B(y,x)\)

  3. Relate \(B(n,m)\) to \(B(n-1,m+1)\). Then calculate the value of \(B(n,m)\) for \(n,m\) natural positives.

  4. Use the result to calculate

    \[ ∫_ 0^{𝜋/2} \sin (t)^ 9\cos (t)^ 7~ dt~ ~ . \]

Hidden solution: [UNACCESSIBLE UUID ’1BD’] [1BF] Prerequisites:convex functions.Let \(I⊂ℝ\) be an open interval, and \(x_ 0∈ I\). Prove that these two facts are equivalent:

  1. \(F:I→ℝ\) is convex.

  2. There exists \(f:I→ℝ\) monotonic (weakly) increasing, and such that \(F(x)=F(x_ 0)+∫_{x_ 0}^ x f(s) \, {\mathbb {d}}s\),

and verify that you can choose \(f\) be the right (or left) derivative of \(F\). [1BG]Exhibit an integrable function \(f :[0,1] →ℝ\) such that the derivative of the function \(F (x) = ∫_ a^ x f (t)dt\) is not \(f\). Hidden solution: [UNACCESSIBLE UUID ’1BH’] [1BJ]Calculate explicitly 1 primitive formulas for

\[ \frac 1{\sin (x)^ 2} ~ ~ ,~ ~ \frac{1}{\sqrt{1+x^ 2}} ~ ~ ,~ ~ \frac{1}{2+\sin (x)}\quad . \]

Hidden solution: [UNACCESSIBLE UUID ’1BK’] [1BM]We define the Gamma function \(Γ:(0,∞)→ℝ\) as

\[ Γ(x) = ∫_ 0^∞ t^{x-1}e^{-t}\, {\mathbb {d}}t~ . \]

  • Show that \(Γ(x)\) is well defined for \(x{\gt}0\) real.

  • Show that \(Γ(x+1)=x Γ(x)\) and deduce that \(Γ(n+1)=n!\) for \(n∈ℕ\).

  • Show that \(Γ(x)\) is analytic.

    (You can assume that derivatives of \(Γ\) are \(Γ^{(n)}(x) = ∫_ 0^∞ (\log t)^ n t^{x-1}e^{-t}\, {\mathbb {d}}t\); those are obtained by derivation under integral sign.)

[1BN]Calculate

\[ \lim _{n\to \infty } \frac 1 n + \frac 1{n+1} + \dots +\frac 1{3n} \]

seeing it as an approximate sum of a Riemann integral. [1BP] Prerequisites:3.Let \(a∈ℝ\), let \(I\) be open interval with \(a∈ I\), and \(𝜑_ 0:I→ℝ\) continuous.

We recursively define \(𝜑_ n:I→ℝ\) for \(n≥ 1\) via \(𝜑_{n}(x)=∫_ a^ x 𝜑_{n-1}(t)\, {\mathbb {d}}t\); show that

\begin{equation} 𝜑_{n+1}(x)=\frac 1{n!} ∫_ a^ x (x-t)^ n𝜑_ 0(t)\, {\mathbb {d}}t \label{eq:multi_ primitiva} \end{equation}
374

Hidden solution: [UNACCESSIBLE UUID ’1BQ’] [1BR] Prerequisites:3.Note:See also Apostol [ 3 ] .

Fix \(a∈ℝ\), and \(I\) open interval with \(a∈ I\); assuming that \(f:I→ℝ\) is if class \(C^{n+1}\), prove Taylor’s formula with integral remainder

\[ f(x) = ∑_{k=0}^ n \frac{f^{(k)}(a)}{k!} (x-a)^ k + \frac 1{n!} ∫_ a^ x(x-t)^ nf^{(n+1)}(t) \, {\mathbb {d}}t\quad . \]

Hidden solution: [UNACCESSIBLE UUID ’1BS’] [1BT] Prerequisites:1, 4.Let \(I⊂ ℝ\) be an open interval. Suppose that \(g:I→ℝ\) in Riemann integrable on any bounded closed interval contained in \(I\). Fixed \(x,y\in {\mathbb {R}}\) with \(x≠ y\), let

\[ R(x,y )=\frac 1{y-x}∫_ x^ y g(s)\, {\mathbb {d}}s \]

(with the usual convention that \(∫_ x^ y g(s)\, {\mathbb {d}}s=-∫_ y^ x g(s)\, {\mathbb {d}}s\), so that \(R(x,y)=R(y,x)\)). If \(g\) is monotonic, show that \(R(x,y)\) is monotonic in each variable. If \(g\) is continuous and \(R(x,y)\) is monotonic in each variable, show that \(g\) is monotonic.

Hidden solution: [UNACCESSIBLE UUID ’1BV’] [1BW]Let \(f:[a,b]→ℝ\) continuous and such that

\[ ∫_ a^ b f(x) g(x)\, {\mathbb {d}}x = 0 \]

for any \(g:[a,b]→ℝ\) continuous: prove that \(f≡ 0\). [1BX] Let’s go back to the exercise 3: computing the Cauchy product of the series \(∑_{n=1}^∞ \frac{(-1)^{n-1}}{\sqrt{n}}\) with itself, produces the series \(∑_ n (-1)^ n c_ n\) with \(c_ n = ∑_{k=1}^{n-1} \frac 1{\sqrt{k(n-k)}}\); show that \(c_ n→ 𝜋\).

Hidden solution: [UNACCESSIBLE UUID ’1BY’] [1BZ]Difficulty:*.Suppose that \(f:ℝ→ℝ\) is continuous and bounded, show that

\[ \lim _{y→ 0+} \frac y𝜋 ∫_{-∞}^∞ \frac{f(x)}{x^ 2+y^ 2}\, {\mathbb {d}}x = f(0) \]

(Hint. start with the case when \(f\) is constant.) [1C0]Let \(n,m≥ 1\) be integers, and set

\[ I_{n,m} = ∫_ 0^ 1 x^ n (\log x)^ m\, {\mathbb {d}}x\quad : \]

relate \(I_{n,m}\) with \(I_{n,m-1}\); use that relation to explicitly calculate

\[ ∫_ 0^ 1 x^ n (\log x) ^ n\, {\mathbb {d}}x~ . \]

Hidden solution: [UNACCESSIBLE UUID ’1C1’] [1C2]Prerequisites:374.Difficulty:**.Show identities

\begin{align} ∫_ 0^ 1 x^{-x}\, dx & = ∑_{n=1}^∞ n^{-n}& & (=\sim 1.291285997\dots ) \\ {} ∫_ 0^ 1 x^ x \, dx & = ∑_{n=1}^∞ (-1)^{n+1}n^{-n} & & (=\sim 0.783430510712\dots ) \end{align}

(Hint: use the Taylor series \(e^ z\), and substitute \(z=x\log (x)\); use the exercise 374 above.) [1C3]Difficulty:*.Let \(f:[0,1]→ℝ\) be Riemann integrable and \(𝜑:ℝ→ℝ\) convex: show that

\begin{equation} \label{eq:dis_ Jensen} 𝜑\left( ∫_ 0^ 1f(x))\, {\mathbb {d}}x\right) ≤ ∫_ 0^ 1𝜑(f(x))\, {\mathbb {d}}x \quad . \end{equation}
377

This result is known as Jensen’s inequality. [1C4]Difficulty:*.Suppose that \(f:(0,1)→(0,∞)\) is continuous and decreasing and \(∫_ 0^ 1 f(t)\, {\mathbb {d}}t{\lt}∞\) then \(\lim _{r→ 0}r f(r)=0\).

Other exercises regarding Riemann integration can be found in 6, 2, 5 (part 8).

  1. Taken from the book by Giaquinta and Modica [ 8 ] , p. 162 and following.