Newton–Leibniz Theorem and Techniques of Integration

The Great Theorem

The Newton–Leibniz theorem is the principal result of mathematical analysis. It unites two problems that might at first appear unrelated: finding the area (integral) and finding the derivative (differential).

Statement: If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$ (i.e., $F' = f$), then:

$ \int_a^b f(x), dx = F(b) - F(a) $

This means: to compute the area under a curve, it is enough to find an antiderivative and substitute the endpoints.

First part of the theorem: The function $\Phi(x) = \int_a^x f(t) dt$ is differentiable, and $\Phi'(x) = f(x)$.

In other words: differentiation and integration are inverse operations. The integral of the derivative = the original function (up to an additive constant).

Methods of Integration

1. Table of integrals (we use reverse differentiation formulas):

• $\int x^n dx = \dfrac{x^{n+1}}{n+1} + C$
• $\int e^x dx = e^x + C$
• $\int \sin x dx = -\cos x + C$
• $\int dx/x = \ln|x| + C$

2. Substitution (change of variable): $\int f(g(x))g'(x)dx = \int f(u)du$ for $u = g(x)$.

Example: $\int \sin(x^2)\cdot 2x dx$; let $u = x^2$, $du = 2x dx$: $= \int \sin(u)du = -\cos(u) + C = -\cos(x^2) + C$.

3. Integration by parts: $\int u, dv = uv - \int v, du$.

Mnemonic: LIATE (Logarithms, Inverse trig, Algebraic, Trigonometric, Exponential) — choose $u$.

Example: $\int x e^x dx$. $u = x$, $dv = e^x dx$; $du = dx$, $v = e^x$.

$= x e^x - \int e^x dx = x e^x - e^x + C = (x-1)e^x + C$.

4. Decomposition into partial fractions: for rational functions we break into a sum of simple fractions.

Applications of the Definite Integral

Area: $S = \int_a^b |f(x) - g(x)| dx$ (area between two curves).

Volume of a solid of revolution: $V = \pi\int_a^b [f(x)]^2 dx$ (revolving around the $Ox$ axis).

Arc length: $L = \int_a^b \sqrt{1 + [f'(x)]^2} dx$.

Physical applications: work of a force, moment of inertia, center of mass.

The Newton–Leibniz integral computes all of this by a single method — this is its power.

Additional Integration Techniques

Trigonometric substitutions. For integrals such as $\int \sqrt{a^2-x^2},dx$ set $x = a \sin t$: $\sqrt{a^2-x^2} = a \cos t$, $dx = a \cos t,dt$. We get $\int a^2 \cos^2 t dt = (a^2/2)(t + \sin t \cos t) + C$.

Recurrence formulas. $I_n = \int \sin^n x, dx$ is expressed in terms of $I_{n-2}$: $I_n = -\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n} \cdot I_{n-2}$. This allows every power of sine to be reduced to $I_0 = x$ or $I_1 = -\cos x$.

Decomposition into partial fractions. For $\int dx/((x+1)(x+2))$ decompose: $1/((x+1)(x+2)) = 1/(x+1) - 1/(x+2)$. Integral: $\ln|x+1| - \ln|x+2| + C = \ln\left|\frac{x+1}{x+2}\right| + C$. This is a standard method for rational functions.

Newton–Leibniz Formula and ODEs

The antiderivative and ODEs are closely linked. The problem $y' = f(x), y(x_0) = y_0$ has the solution $y(x) = y_0 + \int_{x_0}^x f(t) dt$ — a direct application of the theorem. This is the simplest case of a “separable variable equation”: when $f$ depends only on $x$, integrating gives an explicit answer.

Geometric Applications

Area in polar coordinates: $S = \frac{1}{2} \int_a^b r(\theta)^2 d\theta.$ Length of a curve in parametric form: $L = \int_a^b \sqrt{\dot{x}^2 + \dot{y}^2} dt.$ Surface area of revolution: $S = 2\pi \int_a^b |f(x)| \sqrt{1 + f'(x)^2} dx.$

All these formulas are concrete applications of the Newton–Leibniz theorem to geometric problems.

Integration by Parts in Probability Theory

The integration by parts formula $\int u, dv = uv - \int v, du$ has an important application in probability theory. Mathematical expectation $E[X] = \int_0^\infty P(X > t), dt$ for a non-negative random variable $X$ — this is integration by parts as applied to the formula $E[X] = \int_0^\infty x, f(x), dx$ with $u = x, dv = f(x)dx$. This connection between expectation and the distribution function is used in actuarial mathematics and reliability theory to compute mean lifetime and mean time to failure.

Question for reflection: The function $f(x) = x \sin(1/x)$ for $x \ne 0$ and $f(0) = 0$ is continuous. But does there exist an antiderivative $F$ such that $F' = f$? Does every continuous function always have an antiderivative?

Mean Value Theorem for the Definite Integral

First mean value theorem: if $f$ is continuous on $[a, b]$, then there exists $c \in (a, b)$ such that $\int_a^b f(x) dx = f(c)(b-a)$. Geometrically: the area under the curve equals the area of a rectangle with the “average” value of the function. Second mean value theorem: if $f$ is monotonic and $g$ is continuous, then $\int_a^b f(x)g(x) dx = f(a)\int_a^c g dx + f(b)\int_c^b g dx$ for some $c$. In probability theory, the first mean value theorem means: mathematical expectation $E[f(X)] = f(c)$ for some $c$ — there exists a point where the function takes its “average” value.