Integration Techniques

Many integration formulae may be obtained from their corresponding derivative formulas. However, some integration difficulties need additional study. Substitution and variable change, integration by parts, trigonometric integrals and trigonometric replacements take additional study. Let’s look at the different integration techniques in mathematics.

Substitution

Integration by substitution is a helpful technique of integration that is employed when the function to be integrated is either complicated or direct integration is not possible. This technique of integration by substitution simplifies the integral of a function by converting the provided function to a simpler function.

When the integration of a given function is not direct because the algebraic function is not in the standard form, integration by substitution is utilised. Furthermore, proper substitutions can reduce the provided function to its standard form.

Powers of Sine and Cosine

Functions comprising sine and cosine products may be integrated using substitution and trigonometric identities. These might be time-consuming at times, but the approach is simple.

For example:

∫sin5𝑥𝑑𝑥 

=∫sin𝑥sin4𝑥𝑑𝑥

=∫sin𝑥(sin2𝑥)2𝑑𝑥

=∫sin𝑥(1−cos2𝑥)2𝑑𝑥

Using 𝑢=cos𝑥, 𝑑𝑢=−sin𝑥𝑑𝑥

∫sin𝑥(1−cos2𝑥)2𝑑𝑥

=∫−(1−𝑢2)2𝑑𝑢

=∫−(1−2𝑢2+𝑢4)𝑑𝑢

=−𝑢+(⅔)𝑢3−15𝑢5+𝐶

=−cos𝑥+(⅔)cos3𝑥−15cos5𝑥+𝐶

Trigonometric Substitutions

Trigonometric substitution is the replacement of trigonometric functions for other expressions in mathematics. Trigonometric substitution is a method for assessing integrals in calculus. Furthermore, the trigonometric identities may simplify some integrals, including radical expressions.

Trigonometric replacements, like other substitutions in calculus, are a way for assessing an integral by reducing it to a simpler one. Trigonometric replacements employ patterns in the integrand that mimic typical trigonometric relations. They are most commonly effective for integrals of radical or rational functions not easily assessed using other approaches. These substitutions are often combined with fundamental trigonometric relations, product-to-sum identities, and other integration methods like integration by parts and u-substitutions.

Integration by Parts

Integration by parts is a technique for combining the results of two or more functions. The two to be integrated functions, f(x) and g(x), are of the form ∫f (x). g(x). As a result, it is a product rule of integration. The first function, f(x), is chosen such that its derivative formula exists, and the second function, g(x), is chosen so that an integral of such a function exists.

∫f(x).g(x).dx = f(x)∫g(x).dx – ∫(f'(x)∫g(x).dx).dx + C

(First Function x Second Function) integration = (First Function) x (Integration of Second Function) – Integration of (Differentiation of First Function x Integration of Second Function).

Rational Functions

This approach converts the integral of a difficult rational function into the sum of integrals of simpler functions. The following may be found in the denominators of partial fractions:

• non-repeated linear factors
• repeated linear factors
• non-repeated irreducible quadratic factors
• repeated irreducible quadratic factors

Numerical Integration

In analysis, numerical integration refers to a large family of techniques for determining the numerical value of a definite integral. It is also used to describe the numerical solution of differential equations.

The phrase numerical quadrature (commonly reduced to quadrature) is synonymous with numerical integration, mainly used for one-dimensional integrals. Some writers refer to numerical integration as cubature in more than one dimension, while others include higher-dimensional integration in quadrature.

The fundamental goal in numerical integration is to obtain an approximate solution to a specified integral with a particular degree of precision. Several approaches estimate the integral to the necessary accuracy if f(x) is a smooth function integrated across a finite number of dimensions and the domain of integration is bounded.

Partial Functions

A rational function is a ratio of two polynomials P(x)/Q(x), where Q(x) ≠ 0. If the degree of P(x) is smaller than of Q(x), it is a proper fraction; otherwise, it is an improper fraction. If a fraction is improper, the long division method can reduce it to a proper fraction.

P(x)/Q(x) = T(x) + P1(x)/Q(x), where T(x) is a polynomial and P1(x)/Q(x) is a proper rational fraction, if P(x)/Q(x) is an improper fraction. We partition fractions into partial fractions because it makes doing certain integrals easier. It is also used in the Laplace transform.

We can only conduct partial fractions if the numerator’s degree is smaller than the denominator’s degree. As a result, after you’ve realized that partial fractions are conceivable, you’ll need to factor the denominator as thoroughly as possible.

Conclusion

Integration is a way of adding values on a large scale when ordinary addition cannot be performed. However, there are other ways of integration used in Mathematics to integrate functions. Several integration techniques can determine an integral of a function, which makes evaluating the original integral simpler.