Integral

Let \(P\) be a partition of a segment \([a, b]\):

\[ a = x_0 < x_1 < \ldots < x_{n-1} < x_n = b, \]

and \(\xi_k \in [x_{k-1}, x_k]\), \(k=1, \ldots, n\).

Denote \(d(P) = \max\limits_{1 \leqslant k \leqslant n} (x_k - x_{k-1})\). Riemann integral of a function \(f \in C[a, b]\) is a limit of Riemann sums

\[ \int\limits_a^b f(x)\,dx = \lim\limits_{d(P) \to 0} \sum\limits_{k=1}^n f(\xi_k)(x_k - x_{k-1}). \]

Each summand in the Riemann sum represents the area of a thin rectangle with height \(f(\xi_k)\). Therefore, the integral \(\int\limits_a^b f(x)\,dx\) equals area under the graph \(y = f(x)\).

Rules of integration

Integraion by parts: if \(f, g \in C^1[a,b]\) then

\[ \int\limits_a^b f(x) g'(x)\, dx = \left.f(x)g(x)\right|_a^b - \int\limits_a^b f'(x) g(x)\, dx. \]

Change of variable: if \(\varphi \in C^1[a, b]\) is a monotonic function, \(\varphi([\alpha, \beta]) = [a, b]\), then

\[ \int\limits_a^b f(x) \, dx = \int\limits_\alpha^\beta f(\varphi(t)) \varphi'(t)\, dt. \]

Fundamental theorem of calculus

If \(f' \in C[a, b]\) then

\[ \int\limits_a^b f'(x)\,dx = f(b) - f(a). \]

In other words,

\[ \int\limits_a^b g(x)\,dx = G(b) - G(a) \]

where \(G\) is antiderivative of \(g\).

Improper integrals

If \(f \in C[a, b]\) for all \(b > a\) then improper integral over \([a, +\infty)\) is defined as

\[ \int\limits_a^{+\infty} f(x)\,dx = \lim\limits_{b\to +\infty}\int\limits_a^b f(x)\,dx \]

if this limit does exist. In such case improper integral is called convergent, otherwise — divergent.

Fundamental theorem of calculus also works for convergent integrals. For example,

\[ \int\limits_0^{+\infty} e^{-x}\, dx = \left. -e^{-x} \right|_0^{+\infty} = 1 - 0 = 1 \]

(here we use property \(e^{-\infty} \equiv \lim\limits_{t\to +\infty} e^{-t} = 0\).)

Gamma function

The gamma function \(\Gamma(\alpha)\), \(\alpha > 0\), is defined as

\[ \Gamma(\alpha) = \int\limits_0^{+\infty} x^{\alpha -1} e^{-x}\,dx. \]

Properties of gamma function:

• \(\Gamma(n) = (n-1)!\) if \(n\in\mathbb N\)

• \(\Gamma(\alpha + 1) = \alpha \Gamma(\alpha)\) if \(\alpha > 0\)

• \(\Gamma(\alpha) \Gamma(1 - \alpha) = \frac \pi{\sin\pi\alpha}\) if \(0 < \alpha < 1\) (complement formula)

• Gamma function is positive and stricty convex

• Gamma function is infinitely differentiable and

  \[ \Gamma^{(n)}(\alpha) = \int\limits_0^{+\infty} x^{\alpha -1} e^{-x}\ln^n x\,dx \]

The log derivative of gamma function is called digamma function:

\[ \psi(\alpha) = (\log\Gamma(\alpha))' = \frac{\Gamma'(\alpha)}{\Gamma(\alpha)}. \]

Digamma function also can be represented as

\[ \psi(\alpha) = -\gamma + \int\limits_0^1 \frac{1 - t^{\alpha -1}}{1-t}\, dt = -\gamma + \sum\limits_{n=1}^{\infty}\Big(\frac 1n - \frac 1{n + \alpha - 1}\Big) \]

where \(\gamma\) is Euler—Mascheroni constant.

Beta function

The beta function is defined as

\[ B(p, q) = \int\limits_0^1 x^{p-1} (1-x)^{q-1}\,dx = \int\limits_0^{+\infty} \frac{t^{p-1}\,dt}{(1 + t)^{p + q}}, \quad p, q > 0. \]

Beta function is symmetric: \(B(p, q) = B(q, p)\). Also, it can be expressed in terms of gamma function:

\[ B(p, q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p + q)}. \]

Exercises

1. Calculate \(\int\limits_0^1 \ln x\,dx\).

2. Prove that \(\Gamma(n) = (n-1)!\) if \(n\in\mathbb N\).

3. Using the complement formula, find \(\Gamma(1/2)\).

4. Show that \(\Gamma(n + \frac 12) = \frac{(2n-1)!!}{2^n}\sqrt \pi\) if \(n\in \mathbb N\).

5. Calculate the Poisson integral \(\int\limits_0^{+\infty} e^{-x^2}\, dx\).

6. Find \(\Gamma'(1)\) using expansion of digamma function.

7. Calculate \(\int\limits_0^{\frac \pi 2} \sin^n x\,dx\).