Chapter 5

Let \(y\) be any real number and \(b>0 .\) Define \(f:(0, \infty) \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) as, \(f(x):=x^{y}\) and \(g(x):=b^{x} .\) Show that \(f\) and \(g\) are differentiable and find their derivative.

Let \(f\) be in \(\mathscr{R}[a, b] .\) Prove that \(-f\) is in \(\mathscr{R}[a, b]\) and $$ \int_{a}^{b}-f(x) d x=-\int_{a}^{b} f(x) d x . $$

Let \(f:[0,1] \rightarrow \mathbb{R}\) be defined by \(f(x):=x^{3}\) and let \(P:=\\{0,0.1,0.4,1\\} .\) Compute \(L(P, f)\) and \(U(P, f)\)

Compute \(\frac{d}{d x}\left(\int_{-x}^{x} e^{s^{2}} d s\right)\).

Let \(b>0, b \neq 1\) be given. a) Show that for every \(y>0,\) there exists a unique number \(x\) such that \(y=b^{x} .\) Define the logarithm base \(b\), \(\log _{b}:(0, \infty) \rightarrow \mathbb{R},\) by \(\log _{b}(y):=x\) b) Show that \(\log _{b}(x)=\frac{\ln (x)}{\ln (b)}\). c) Prove that if \(c>0, c \neq 1,\) then \(\log _{b}(x)=\frac{\log _{c}(x)}{\log _{c}(b)}\) d) Prove \(\log _{b}(x y)=\log _{b}(x)+\log _{b}(y),\) and \(\log _{b}\left(x^{y}\right)=y \log _{b}(x)\).

Let \(f\) and \(g\) be in \(\mathscr{R}[a, b] .\) Prove, without using Proposition \(5.2 .5,\) that \(f+g\) is in \(\mathscr{R}[a, b]\) and $$ \int_{a}^{b}(f(x)+g(x)) d x=\int_{a}^{b} f(x) d x+\int_{a}^{b} g(x) d x $$

Find out for which \(a \in \mathbb{R}\) does \(\sum_{n=1}^{\infty} e^{a n}\) converge. When the series converges, find an upper bound for the sum.

Let \(f:[0,1] \rightarrow \mathbb{R}\) be defined by \(f(x):=x .\) Show that \(f \in \mathscr{A}[0,1]\) and compute \(\int_{0}^{1} f\) using the definition of the integral (but feel free to use the propositions of this section).

Compute \(\frac{d}{d x}\left(\int_{0}^{x^{2}} \sin \left(s^{2}\right) d s\right)\).

(requires \$4.3): Use Taylor's theorem to study the remainder term and show that for all \(x \in \mathbb{R}\) $$ e^{x}=\sum_{n=0}^{\infty} \frac{x^{n}}{n !} $$ Hint: Do not differentiate the series term by term (unless you would prove that it works).

Let \(f:[a, b] \rightarrow \mathbb{R}\) be Riemann integrable. Let \(g:[a, b] \rightarrow \mathbb{R}\) be a function such that \(f(x)=\) \(g(x)\) for all \(x \in(a, b) .\) Prove that \(g\) is Riemann integrable and that $$ \int_{a}^{b} g=\int_{a}^{b} f $$

a) Compute \(p . v \cdot \int_{-1}^{1} 1 / x d x\). b) Compute \(\lim _{\varepsilon \rightarrow 0^{+}}\left(\int_{-1}^{-\varepsilon} 1 / x d x+\int_{2 \varepsilon}^{1} 1 / x d x\right)\) and show it is not equal to the principal value. c) Show that if \(f\) is integrable on \([a, b]\), then \(p . v \cdot \int_{a}^{b} f=\int_{a}^{b} f\) (for an arbitrary \(\left.c \in(a, b)\right)\). d) Suppose \(f:[-1,1] \rightarrow \mathbb{R}\) is an odd function \((f(-x)=-f(x)),\) that is integrable on \([-1,-\varepsilon]\) and \([\varepsilon, 1]\) for all \(\varepsilon>0 .\) Prove that \(p . v \cdot \int_{-1}^{1} f=0\) e) Suppose \(f:[-1,1] \rightarrow \mathbb{R}\) is continuous and differentiable at \(0 .\) Show that p.v. \(\int_{-1}^{1} \frac{f(x)}{x} d x\) exists.

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a bounded function. Suppose there exists a sequence of partitions \(\\{P\\}\) of \([a, b]\) such that $$ \lim _{k \rightarrow \infty}\left(U\left(P_{k}, f\right)-L\left(P_{k}, f\right)\right)=0 $$ Show that \(f\) is Riemann integrable and that $$ \int_{a}^{b} f=\lim _{k \rightarrow \infty} U\left(P_{k}, f\right)=\lim _{k \rightarrow \infty} L\left(P_{k}, f\right) $$

Suppose \(F:[a, b] \rightarrow \mathbb{R}\) is continuous and differentiable on \([a, b] \backslash S,\) where \(S\) is a finite set. Suppose there exists an \(f \in \mathscr{R}[a, b]\) such that \(f(x)=F^{\prime}(x)\) for \(x \in[a, b] \backslash S .\) Show that \(\int_{a}^{b} f=F(b)-F(a)\).

Use the geometric sum formula to show (for \(t \neq-1)\) $$ 1-t+t^{2}-\cdots+(-1)^{n} t^{n}=\frac{1}{1+t}-\frac{(-1)^{n+1} t^{n+1}}{1+t} $$ Using this fact show $$ \ln (1+x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^{n}}{n} $$ for all \(x \in(-1,1]\) (note that \(x=1\) is included). Finally, find the limit of the alternating harmonic series $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}=1-1 / 2+1 / 3-1 / 4+\cdots $$

Prove the mean value theorem for integrals. That is, prove that if \(f:[a, b] \rightarrow \mathbb{R}\) is continuous, then there exists a \(c \in[a, b]\) such that \(\int_{a}^{b} f=f(c)(b-a) .\)

Prove $$ \int_{-\infty}^{\infty}|\operatorname{sinc}(x)| d x=\infty. $$ Hint: Again, it is enough to show this on just one side.

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a continuous function. Let \(c \in[a, b]\) be arbitrary. Define $$F(x):=\int_{c}^{x} f$$ Prove that \(F\) is differentiable and that \(F^{\prime}(x)=f(x)\) for all \(x \in[a, b]\).

Show $$ e^{x}=\lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n} $$ Hint: Take the logarithm. Note: The expression \(\left(1+\frac{x}{n}\right)^{n}\) arises in compound interest calculations. It is the amount of money in a bank account after 1 year if 1 dollar was deposited initially at interest \(x\) and the interest was compounded n times during the year. The exponential \(e^{x}\) is the result of continuous compounding.

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a continuous function such that \(f(x) \geq 0\) for all \(x \in[a, b]\) and \(\int_{a}^{b} f=0 .\) Prove that \(f(x)=0\) for all \(x\).

Can you interpret $$ \int_{-1}^{1} \frac{1}{\sqrt{|x|}} d x $$ as an improper integral? If so, compute its value.

Prove integration by parts. That is, suppose \(F\) and \(G\) are continuously differentiable functions on \([a, b] .\) Then prove $$\int_{a}^{b} F(x) G^{\prime}(x) d x=F(b) G(b)-F(a) G(a)-\int_{a}^{b} F^{\prime}(x) G(x) d x.$$

a) Prove that for \(n \in \mathbb{N}\) we have $$ \sum_{k=2}^{n} \frac{1}{k} \leq \ln (n) \leq \sum_{k=1}^{n-1} \frac{1}{k} $$ b) Prove that the limit $$ \gamma:=\lim _{n \rightarrow \infty}\left(\sum_{k=1}^{n} \frac{1}{k}-\ln (n)\right) $$ exists. This constant is known as the Euler-Mascheroni constant*. It is not known if this constant is rational or not. It is approximately \(\gamma \approx 0.5772\).

Take \(f:[0, \infty) \rightarrow \mathbb{R}\), Riemann integrable on every interval \([0, b],\) and such that there exist \(M,\) \(a,\) and \(T\), such that \(|f(t)| \leq M e^{a t}\) for all \(t \geq T\). Show that the Laplace transform of \(f\) exists. That is, for every \(s>a\) the following integral converges: $$ F(s):=\int_{0}^{\infty} f(t) e^{-s t} d t. $$

Let \(c \in(a, b)\) and let \(d \in \mathbb{R}\). Define \(f:[a, b] \rightarrow \mathbb{R}\) as $$ f(x):=\left\\{\begin{array}{ll} d & \text { if } x=c \\ 0 & \text { if } x \neq c \end{array}\right. $$ Prove that \(f \in \mathscr{A}[a, b]\) and compute \(\int_{a}^{b} f\) using the definition of the integral (but feel free to use the propositions of this section).

Suppose \(F\) and \(G\) are continuously" differentiable functions defined on \([a, b]\) such that \(F^{\prime}(x)=G^{\prime}(x)\) for all \(x \in[a, b] .\) Using the fundamental theorem of calculus, show that \(F\) and \(G\) differ by \(a\) constant. That is, show that there exists a \(C \in \mathbb{R}\) such that \(F(x)-G(x)=C\). The next exercise shows how we can use the integral to "smooth out" a non- differentiable function.

Show $$ \lim _{x \rightarrow \infty} \frac{\ln (x)}{x}=0 $$

Let \(f:[a, b] \rightarrow \mathbb{R}\) and \(g:[a, b] \rightarrow \mathbb{R}\) be continuous functions such that \(\int_{a}^{b} f=\int_{a}^{b}\) g. Show that there exists a \(c \in[a, b]\) such that \(f(c)=g(c)\).

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a Riemann integrable function on every interval \([a, b],\) and such that \(\int_{-\infty}^{\infty}|f(x)| d x<\infty .\) Show that the Fourier sine and cosine transforms exist. That is, for every \(\omega \geq 0\) the following integrals converge $$ F^{s}(\omega):=\frac{1}{\pi} \int_{-\infty}^{\infty} f(t) \sin (\omega t) d t, \quad F^{c}(\omega):=\frac{1}{\pi} \int_{-\infty}^{\infty} f(t) \cos (\omega t) d t. $$ Furthermore, show that \(F^{s}\) and \(F^{c}\) are bounded functions.

Suppose \(f:[a, b] \rightarrow \mathbb{R}\) is Riemann integrable. Let \(\varepsilon>0\) be given. Then show that there exists a partition \(P=\left\\{x_{0}, x_{1}, \ldots, x_{n}\right\\}\) such that if we pick any set of numbers \(\left\\{c_{1}, c_{2}, \ldots, c_{n}\right\\}\) with \(c_{k} \in\left[x_{k-1}, x_{k}\right]\) for all \(k,\) then $$ \left|\int_{a}^{b} f-\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x_{k}\right|<\varepsilon $$

Show that \(e^{x}\) is convex, in other words, show that if \(a \leq x \leq b\), then \(e^{x} \leq e^{a} \frac{b-x}{b-a}+e^{b} \frac{x-a}{b-a}\).

Let \(f \in \mathscr{R}[a, b] .\) Let \(\alpha, \beta, \gamma\) be arbitrary numbers in \([a, b]\) (not necessarily ordered in any way). Prove $$ \int_{\alpha}^{\gamma} f=\int_{\alpha}^{\beta} f+\int_{\beta}^{\gamma} f $$ Recall what \(\int_{a}^{b} f\) means if \(b \leq a\)

Suppose \(f:[0, \infty) \rightarrow \mathbb{R}\) is Riemann integrable on every interval \([0, b] .\) Show that \(\int_{0}^{\infty} f\) converges if and only if for every \(\varepsilon>0\) there exists an \(M\) such that if \(M \leq a

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a Riemann integrable function. Let \(\alpha>0\) and \(\beta \in \mathbb{R}\). Then define \(g(x):=f(\alpha x+\beta)\) on the interval \(I=\left[\frac{a-\beta}{\alpha}, \frac{b-\beta}{\alpha}\right] .\) Show that \(g\) is Riemann integrable on \(I .\)

Suppose \(f:[a, b] \rightarrow \mathbb{R}\) is continuous and \(\int_{a}^{x} f=\int_{x}^{b} f\) for all \(x \in[a, b] .\) Show that \(f(x)=0\) for all \(x \in[a, b] .\)

Using the logarithm find $$ \lim _{n \rightarrow \infty} n^{1 / n} $$

Suppose \(f:[0, \infty) \rightarrow \mathbb{R}\) is nonnegative and decreasing. Prove: a) If \(\int_{0}^{\infty} f<\infty,\) then \(\lim _{x \rightarrow \infty} f(x)=0\). b) The converse does not hold.

Suppose \(f:[0,1] \rightarrow \mathbb{R}\) and \(g:[0,1] \rightarrow \mathbb{R}\) are such that for all \(x \in(0,1]\) we have \(f(x)=g(x) .\) Suppose \(f\) is Riemann integrable. Prove \(g\) is Riemann integrable and \(\int_{0}^{1} f=\int_{0}^{1} g .\)

Suppose \(f:[a, b] \rightarrow \mathbb{R}\) is continuous and \(\int_{a}^{x} f=0\) for all rational \(x\) in \([a, b] .\) Show that \(f(x)=0\) for all \(x \in[a, b]\).

Show that \(E(x)=e^{x}\) is the unique continuous function such that \(E(x+y)=E(x) E(y)\) and \(E(1)=e\). Similarly, prove that \(L(x)=\ln (x)\) is the unique continuous function defined on positive \(x\) such that \(L(x y)=L(x)+L(y)\) and \(L(e)=1\)

Suppose \(f:[a, b] \rightarrow \mathbb{R}\) is bounded and has finitely many discontinuities. Show that as a function of \(x\) the expression \(|f(x)|\) is bounded with finitely many discontinuities and is thus Riemann integrable. Then show $$ \left|\int_{a}^{b} f(x) d x\right| \leq \int_{a}^{b}|f(x)| d x $$

Find an example of an unbounded continuous function \(f:[0, \infty) \rightarrow \mathbb{R}\) that is nonnegative and such that \(\int_{0}^{\infty} f<\infty .\) Note that this means that \(\lim _{x \rightarrow \infty} f(x)\) does not exist; compare previous exercise. Hint: On each interval \([k, k+1], k \in \mathbb{N}\), define a function whose integral over this interval is less than say \(2^{-k}\).

Let \(f:[0,1] \rightarrow \mathbb{R}\) be a bounded function. Let \(P_{n}=\left\\{x_{0}, x_{1}, \ldots, x_{n}\right\\}\) be a uniform partition of \([0,1],\) that is, \(x_{j}:=j / n .\) Is \(\left\\{L\left(P_{n}, f\right)\right\\}_{n=1}^{\infty}\) always monotone? Yes/No: Prove or find a counterexample.

A function \(f\) is an odd function if \(f(x)=-f(-x),\) and \(f\) is an even function if \(f(x)=f(-x)\) Let \(a>0 .\) Assume \(f\) is continuous. Prove: a) If \(f\) is odd, then \(\int_{-a}^{a} f=0\). b) If \(f\) is even, then \(\int_{-a}^{a} f=2 \int_{0}^{a} f\).

(requires \(\$ 4.3):\) Since \(\left(e^{x}\right)^{\prime}=e^{x},\) it is easy to see that \(e^{x}\) is infinitely differentiable (hasderivatives of all orders). Define the function \(f: \mathbb{R} \rightarrow \mathbb{R}\). $$ f(x):=\left\\{\begin{array}{ll} e^{-1 / x} & \text { if } x>0 \\ 0 & \text { if } x \leq 0 \end{array}\right. $$ a) Prove that for any \(m \in \mathbb{N}\), $$ \lim _{x \rightarrow 0^{+}} \frac{e^{-1 / x}}{x^{m}}=0 $$ b) Prove that \(f\) is infinitely differentiable. c) Compute the Taylor series for \(f\) at the origin, that is, $$ \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k !} x^{k} $$ Show that it converges, but show that it does not converge to \(f(x)\) for any \(x>0 .\)

(More challenging): Find an example of a function \(f:[0, \infty) \rightarrow \mathbb{R}\) integrable on all intervals such that \(\lim _{n \rightarrow \infty} \int_{0}^{n} f\) converges as a limit of a sequence (so \(\left.n \in \mathbb{N}\right),\) but such that \(\int_{0}^{\infty} f\) does not exist. Hint: For all \(n \in \mathbb{N}\), divide \([n, n+1]\) into two halves. In one half make the function negative, on the other make the function positive.

a) Show that \(f(x):=\sin (1 / x)\) is integrable on any interval (you can define \(f(0)\) to be anything). b) Compute \(\int_{-1}^{1} \sin (1 / x) d x\). (Mind the discontinuity)

When a function \(f\) can be written as $$ f(x)=\sum_{k=1}^{n} \alpha_{k} \varphi_{I_{k}}(x) $$ for some real numbers \(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}\) and some bounded intervals \(I_{1}, I_{2}, \ldots, I_{n},\) then \(f\) is called a step function.Let \(I\) be an arbitrary bounded interval (you should consider all types of intervals: closed, open, half-open) and \(a

Suppose \(f:[1, \infty) \rightarrow \mathbb{R}\) is such that \(g(x):=x^{2} f(x)\) is a bounded function. Prove that \(\int_{1}^{\infty} f\) converges. It is sometimes desirable to assign a value to integrals that normally cannot be interpreted as even improper integrals, e.g. \(\int_{-1}^{1} 1 / x d x .\) Suppose \(f:[a, b] \rightarrow \mathbb{R}\) is a function and \(a0 .\) Define the Cauchy principal value of \(\int_{a}^{b} f\) as $$ \text { p.v. } \int_{a}^{b} f:=\lim _{\varepsilon \rightarrow 0^{+}}\left(\int_{a}^{c-\varepsilon} f+\int_{c+\varepsilon}^{b} f\right) \text { , } $$ if the limit exists.

Let \(f:[a, b] \rightarrow \mathbb{R}\) be increasing. a) Show that \(f\) is Riemann integrable. Hint: Use a uniform partition; each subinterval of same length. b) Use part a to show that a decreasing function is Riemann integrable. c) Suppose* \(h=f-g\) where \(f\) and g are increasing functions on \([a, b] .\) Show that h is Riemann integrable.