Integration in Calculus: Techniques, Applications, and Examples

Integral
The integral is a fundamental concept in calculus that deals with the accumulation of infinitesimal quantities. It is used to compute areas, volumes, and other quantities that are related to the accumulation of values. There are two main types of integrals: definite and indefinite integrals. Integrals are closely related to the concept of a derivative, and together they form the two main building blocks of calculus.

Indefinite Integral (Antiderivative):
An indefinite integral, also known as an antiderivative, represents a family of functions whose derivatives are the same. Given a function $f(x)$, the antiderivative or indefinite integral of $f(x)$ is represented as:
$\int f(x) \, dx =F(x)+C$
Here, $F(x)$ is the antiderivative of $f(x)$, $dx$ indicates that the integration is with respect to the variable $x$, and $C$ is the constant of integration, representing the family of functions that have the same derivative.
For example, the indefinite integral of $f(x)=x$ would be: $\int xdx= \frac{1}{2} x^2 + C$

Definite Integral:
The definite integral, on the other hand, represents the net accumulation of a quantity between two points in a domain. It can be thought of as the signed area under a curve of a function $f(x)$ between two points $a$ and $b$. The definite integral is represented as:
$\int_{a}^{b} f(x) \, dx$

The Fundamental Theorem of Calculus (FTC) connects the concepts of derivatives and integrals. It states that if a function $f(x)$ is continuous on the interval $[a,b]$ and $F(x)$ is an antiderivative of $f(x)$, then:
$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$

This theorem allows us to compute definite integrals u\sing antiderivatives.

For example, let's compute the definite integral of $f(x)=x$ from 0 to 2:
$\int_{0}^{2} x \, dx = \left[\frac{1}{2}x^2\right]_{0}^{2} = \frac{1}{2}(2)^2 - \frac{1}{2}(0)^2 = 2$
In this case, the definite integral represents the area under the curve $y=x$ between $x=0$ and $x=2$

Integration Techniques:
There are several techniques to compute integrals, including substitution, integration by parts, partial fractions, and trigonometric substitution. These techniques are used to simplify or break down more complex integrals into easier-to-handle expressions that can be integrated directly or in terms of known antiderivatives.

Substitution (u-substitution):
Substitution is a technique used to simplify integrals by transforming them into a new variable. It involves choo\sing a substitution $u$ and its corresponding differential du such that the integral becomes easier to solve. The general steps are:
Choose a substitution $u=g(x)$
Compute the differential $du=g' (x)dx$
Replace $x$ and $dx$ in the original integral with the substitution and the differential.
Solve the new integral in terms of $u$
Substitute $u$ back in terms of $x$ to obtain the final result.

Example: $\int x \cdot e^{x^2 } dx$
We can choose the substitution $u=x^2$. Then, we compute the differential $du = 2x dx$. Now, we can rewrite the integral in terms of $u$:
$\int \frac{1}{2} e^u du$
Now, we can integrate with respect to $u$:
$\frac{1}{2} \int e^u du = \frac{1}{2} e^u + C$
Finally, we substitute $u$ back in terms of $x$: $\frac{1}{2} e^{x^2 } +C$

Integration by Parts:
Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation. The formula for integration by parts is:
$\int udv=uv- \int vdu$. Here, $u$ and $v$ are functions of $x$. To use this method, you need to choose $u$ and $dv$ from the integrand and then compute $du$ and $v$.

Example: $\int xe^x dx$
We can choose $u = x$ and $dv=e^x dx$. Then, we compute $du = dx$ and $v=e^x$. Applying the integration by parts formula:
$\int xe^x dx =xe^x - \int e^x dx$
Now, we integrate $e^x$: $xe^x-e^x+C$

Partial Fractions:
Partial fractions is a technique used to integrate rational functions (fractions with polynomials in the numerator and denominator). The method involves decompo\sing the rational function into simpler fractions with linear or quadratic denominators. These simpler fractions are easier to integrate.

To perform partial fraction decomposition, first ensure that the degree of the numerator is less than the degree of the denominator. If it's not, perform polynomial division. Then, decompose the rational function into its simpler fractions u\sing algebraic techniques.

Example: $\int \frac{1}{x^2 - 1} dx$
We can decompose the rational function u\sing partial fractions:
$\frac{1}{x^2 - 1} = \frac{A}{x - 1} + \frac{B}{x + 1}$

Solving for $A$ and $B$, we find that $A= \frac{1}{2}$ and $B =-\frac{1}{2}$.

So,$\int \frac{1}{x^2 - 1} \, dx = \int \left(\frac{1}{2} \cdot \frac{1}{x - 1} - \frac{1}{2} \cdot \frac{1}{x + 1}\right) \, dx$

Now, we can integrate the simpler fractions:
$\int \frac{1}{x^2 - 1} \, dx = \frac{1}{2} \int \frac{1}{x - 1} \, dx - \frac{1}{2} \int \frac{1}{x + 1} \, dx$

Integrating each term, we get:
$\frac{1}{2} \ln |x-1| - \frac{1}{2} \ln |x+1| + C$

We can also combine the logarithms:
$\int \frac{1}{x^2 - 1} \, dx = \frac{1}{2} \ln \left| \frac{x - 1}{x + 1} \right| + C$

Trigonometric Substitution:
Trigonometric substitution is a technique used to integrate expressions involving square roots of quadratic functions. It involves substituting a trigonometric function for the variable in the integrand, which simplifies the expression and allows for integration. The substitution chosen depends on the form of the expression:
For $\sqrt{a^2-x^2}$, use $x=a \sin \theta$
For $\sqrt{a^2+x^2}$, use $x=a \tan \theta$
For $\sqrt{x^2-a^2}$, use $x=a \sec \theta$

Example: $\int \frac{1}{\sqrt{1-x^2}} \, dx$
We can use the substitution $x= \sin \theta$:
$\int \frac{1}{\sqrt{1 - \sin^2 \theta}} \cos \theta \, d\theta = \int \frac{\cos \theta}{\sqrt{1 - \sin^2 \theta}} \, d\theta = \int \frac{\cos \theta}{\cos \theta} \, d\theta$

Now, we can integrate: $\int 1d \theta = \theta +C$
Finally, we substitute back in terms of $x$: $arc \sin x+C$

These are just a few of the many integration techniques available. Depending on the integrand, one or more of these techniques may be needed to find the integral. In some cases, integrals cannot be expressed in terms of elementary functions and require special functions, such as the error function, or numerical methods for evaluation.

Improper Integrals:
Improper integrals are integrals that involve infinite limits or integrands with discontinuities in the interval of integration. They are not defined in the usual sense but can often be assigned a value through a limit process. There are two types of improper integrals:

Type 1: Infinite limits
$\int_a^\infty f(x) \, dx$
To evaluate this type of improper integral, we take the limit as the upper bound approaches infinity:
$\int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx$

Type 2: Discontinuous integrands
$\int_a^b \frac{1}{x} \, dx$
If there is a discontinuity at c∈[a,b], we split the integral into two parts:
$\int_a^b \frac{1}{x} \, dx = \int_a^c \frac{1}{x} \, dx + \int_c^b \frac{1}{x} \, dx$

Now, we take the limit as the integration points approach the discontinuity:
$\lim_{t \to c^-} \int_a^t \frac{1}{x} \, dx + \lim_{s \to c^+} \int_s^b \frac{1}{x} \, dx$

Multivariable Integration:
Integration can be extended to functions of multiple variables. For example, double integrals involve integrating a function of two variables over a region in the plane:
$\iint_R f(x, y) \, dx \, dy$

To compute a double integral, we usually perform two successive single-variable integrations, one for each variable. The order of integration can sometimes be changed to simplify the computation.

Triple integrals involve integrating a function of three variables over a region in space:
$\iiint_V f(x, y, z) \, dx \, dy \, dz$

Similar to double integrals, triple integrals can be computed by performing three successive \single-variable integrations.

Line Integrals:
Line integrals involve integrating a function along a curve in the plane or space. Given a scalar function $f(x,y)$ and $a$ curve $C$, the line integral is defined as:
$\int_C f(x,y) \, ds$ where $ds$ represents an infinitesimal arc length along the curve.

Line integrals can also be defined for vector fields. Given a vector field $F(x,y)=P(x,y)i+Q(x,y)j$ and $a$ curve $C$, the line integral is defined as:
$\int_C F \cdot dr = \int_C P \, dx + Q dy$

Line integrals are used in various applications, such as computing work done by a force field or the circulation of a fluid around a curve.