Examples on Direct Integration Using Standard Integrals

It is possible to use integration formulas for the integration of algebraic expressions, trigonometric ratios, inverse trigonometric functions, logarithmic, and exponential functions, as well as other functions. The integration of functions yields the original functions for which the derivatives were obtained through differentiation. In order to find the antiderivative of a function, these integration formulas are employed. When we differentiate a function f over an interval I, we obtain a family of functions over the interval. If we know the values of functions in I, we can use that knowledge to determine the function f. Integration is the term used to describe the inverse process of differentiation.

The integration formulas have been presented in the following six sets of formulas as a general overview. Essentially, integration is a method of bringing together parts in order to form a whole. Basic integration formulas, integration of trigonometric ratios, inverse trigonometric functions, the product of functions, and a more advanced set of integration formulas are all included in this collection of resources. Integration is the operation that is the inverse of differentiation. As a result, the fundamental integration formula is ∫ f'(x).dx = f(x) + C.

Basic integration formulas

Indefinite integration is achieved by applying the fundamental theorems of integrals to obtain generalised results, which are referred to as integration formulas in the literature.

• ∫ xn.dx = x(n+1)/(n + 1)+ C
• ∫ 1.dx = x + C
• ∫ ex.dx = ex + C
• ∫1/x.dx = log|x| + C
• ∫ ax.dx = ax /loga+ C

Integration methods

The following are the various methods of integration:

1. Substitution as a method of integration.

2. Integration Using Trigonometric Identities.

3. Integration by Parts.

4. A specific function is integrated into the overall system.

5. Partially Fractional Integration.

Integration by substitution

Sometimes it is extremely difficult to find the integration of a function; as a result, we can find the integration by introducing a new independent variable into the equation. Integration By Substitution is the term used to describe this technique.

By changing the independent variable x to t, the given form of integral function (for example, ∫f(x)) can be transformed into another form of integral function.

Substituting x = g(t) in the function ∫f(x), we get;

dx/dt = g'(t)

or dx = g'(t).dt

Thus, I = ∫f(x).dx = f(g(t)).g'(t).dt

Integration by parts

A special technique is required for integration by parts of a function, in which the integrand function is the multiple of two or more functions, in order to integrate a function.

Consider the integrand function f as an example of an integrand function f(x).g(x)

When expressed mathematically, integration by parts can be represented as;

∫f(x).g(x).dx = f(x).∫g(x).dx–∫(f′(x).∫g(x).dx).dx

This is what it says:

When two functions are multiplied together, the integral of the product of the two functions is equal to (First function * Integral of second function) – integral of [(differentiation of first function) * Integral of second function].

Integration using trigonometric identities

We use trigonometric identities to simplify a function that can be easily integrated when we are integrating a function whose integrand is any kind of trigonometric function, as shown in the following example.

The following are a few examples of trigonometric identities

• sin2x = 1- cos2x /2
• cos2x = 1+ cos2x/2
• sin3x = 3sinx – sin3x/4
• cos3x = 3cosx + cos3x/4

Integration of some particular function

In order to integrate a specific function, some important integration formulae must be understood and applied. These integration formulae can then be used to integrate other functions into a standard form of the integrand. By employing a direct form of integration method, it is simple to discover how these standard integrands are integrated together.

Integration by partial fraction

Knowing that a Rational Number can be expressed in the form of p/q, where p and q are both integers and q not equal to 0, we can see that a Rational Number can be expressed as Additionally, a rational function is defined as the ratio of two polynomials that can be expressed in the form of partial fractions: P(x)/Q(x), where Q(x) is not equal to zero.

Partially fractional fractions can be divided into two types

• Proper partial fraction: When the degree of the numerator is less than the degree of the denominator, the partial fraction is referred to as a correct partial fraction.

• Improper partial fraction: The term “improper partial fraction” refers to a partial fraction in which the numerator’s degree is greater than the degree of the denominator. As a result, the fraction can be broken down into smaller partial fractions that are easier to integrate.

Conclusion

Internal shear, internal moment, rotation, and deflection of a beam are all measured using the direct integration method, which is used in structural analysis. It is possible to use integration formulas for the integration of algebraic expressions, trigonometric ratios, inverse trigonometric functions, logarithmic, and exponential functions, as well as other functions. Essentially, integration is a method of bringing together parts in order to form a whole. A special technique is required for integration by parts of a function, in which the integrand function is the multiple of two or more functions, in order to integrate a function.In order to integrate a specific function, some important integration formulae must be understood and applied. These integration formulae can then be used to integrate other functions into a standard form of the integrand.