maths > integral-calculus

Methods of Integration

what you'll learn...

Overview

»  Understanding Complexity of Integration Methods
functions given as multiplication or division of sub-functions.

→ Integration by Identities ${\Pi }_{i=1}^{n}{g}_{i}\left(x\right)=\sum _{i=1}^{m}{h}_{i}\left(x\right)$${\Pi}_{i = 1}^{n} {g}_{i} \left(x\right) = {\sum}_{i = 1}^{m} {h}_{i} \left(x\right)$

→ Integration by Substitution where $f\left(x\right)=g\left(h\left(x\right)\right)h\prime \left(x\right)dx$

→ Integration by Parts where $\int g\left(x\right)h\left(x\right)dx=g\left(x\right)\int h\left(x\right)dx-\int \left[\int h\left(x\right)dx\right]g\prime \left(x\right)dx$

→ Special cases of Integration by Parts where $\left(g\prime \left(x\right)\right)\prime =-g\left(x\right)$

→ Special cases of Integration by Parts where $g\left(x\right)={e}^{x}$$g \left(x\right) = {e}^{x}$ and $h\left(x\right)=r\left(x\right)+r\prime \left(x\right)$

→ Integration of Division by Polynomial: $\frac{r\left(x\right)}{q\left(x\right)}=t\left(x\right)+\frac{p\left(x\right)}{q\left(x\right)}$$\frac{r \left(x\right)}{q \left(x\right)} = t \left(x\right) + \frac{p \left(x\right)}{q \left(x\right)}$, where order of $p\left(x\right)$$p \left(x\right)$ is smaller than order of $q\left(x\right)$$q \left(x\right)$.

→ Integration by Partial fractions $\frac{p\left(x\right)}{{\Pi }_{i=1}^{n}{g}_{i}\left(x\right)}=\sum _{i=1}^{m}\frac{1}{{h}_{i}\left(x\right)}$$\frac{p \left(x\right)}{{\Pi}_{i = 1}^{n} {g}_{i} \left(x\right)} = {\sum}_{i = 1}^{m} \frac{1}{{h}_{i} \left(x\right)}$ where $\frac{1}{{h}_{i}\left(x\right)}$$\frac{1}{{h}_{i} \left(x\right)}$ is integrable.

gap

We have learned the standard results of anti-derivatives as inverse of derivative. In that few of the expressions are not covered. For example, consider $\int \mathrm{tan}xdx$$\int \tan x \mathrm{dx}$. This is not solved yet. We will solve this problem in due course of this lesson.

The point being, there are several functions for which finding result of integration is not explained yet in this course.

It is also noted here that not all functions have an anti-derivative in the form of a function. For example, $\int {e}^{-{x}^{2}}dx$$\int {e}^{- {x}^{2}} \mathrm{dx}$ is widely used in other scientific areas of studies, but the integral has no function-form. How do we find integral of this? "by numerical integration or a table". Finding such integration is outside the scope of this course. But knowing that there are functions for which integration cannot be worked out as a function and we can use numerical methods, is good for now.

insights

Though there are several functions for which integration cannot be worked out as a function, there are several other forms of functions, that can be integrated with some algebraic manipulations. The objective of this page is to provide some insights to that.

Usually, functions are made of sub-functions of standard forms with operations.

• standard forms of functions: ${x}^{n}$${x}^{n}$, trigonometric functions, inverse trigonometric functions, exponents, and logarithm

Standard operations:

addition of functions $f\left(x\right)+g\left(x\right)$$f \left(x\right) + g \left(x\right)$,

subtraction of functions $f\left(x\right)-g\left(x\right)$$f \left(x\right) - g \left(x\right)$,

multiplication of functions $f\left(x\right)×g\left(x\right)$$f \left(x\right) \times g \left(x\right)$,

division $\frac{f\left(x\right)}{g\left(x\right)}$$f \frac{x}{g} \left(x\right)$,

multiplication or division by a constant $af\left(x\right)$$a f \left(x\right)$,

exponents ${f\left(x\right)}^{a}$$f {\left(x\right)}^{a}$,

roots ${f\left(x\right)}^{\frac{1}{n}}$$f {\left(x\right)}^{\frac{1}{n}}$,

scaling $f\left(ax\right)$$f \left(a x\right)$,

shifting $f\left(x+a\right)$$f \left(x + a\right)$, and

function of function $f\left(g\left(x\right)\right)$$f \left(g \left(x\right)\right)$ In these, addition, subtraction, and multiplication by a constant were covered in algebra of integrals.

$\int \left[af\left(x\right)±g\left(x\right)\right]dx$$\int \left[a f \left(x\right) \pm g \left(x\right)\right] \mathrm{dx}$$=a\int f\left(x\right)dx±\int g\left(x\right)dx$$= a \int f \left(x\right) \mathrm{dx} \pm \int g \left(x\right) \mathrm{dx}$

summary

In this lesson, we will focus on integration methods of functions with multiplication or division of sub-functions. Assume ${g}_{i}\left(x\right)$${g}_{i} \left(x\right)$ is a standard function for which anti-derivative is known. If a function is given as $f\left(x\right)={\Pi }_{i=1}^{n}{g}_{i}\left(x\right)$$f \left(x\right) = {\Pi}_{i = 1}^{n} {g}_{i} \left(x\right)$, that is the function equals multiplication of several sub-functions. How to work out the integral of the function?

To solve this, several patterns and their properties are understood for different cases. Based on the properties, methods to handle the functions to work out the integration are provided. These are

• Integration by Identities ${\Pi }_{i=1}^{n}{g}_{i}\left(x\right)=\sum _{i=1}^{m}{h}_{i}\left(x\right)$${\Pi}_{i = 1}^{n} {g}_{i} \left(x\right) = {\sum}_{i = 1}^{m} {h}_{i} \left(x\right)$

• Integration by Substitution where $f\left(x\right)=g\left(h\left(x\right)\right)h\prime \left(x\right)dx$

• Integration by Parts where $\int g\left(x\right)h\left(x\right)dx=g\left(x\right)\int h\left(x\right)dx-\int \left[\int h\left(x\right)dx\right]g\prime \left(x\right)dx$

• Special cases of Integration by Parts where $\left(g\prime \left(x\right)\right)\prime =-g\left(x\right)$

• Special cases of Integration by Parts where $g\left(x\right)={e}^{x}$$g \left(x\right) = {e}^{x}$ and $h\left(x\right)=r\left(x\right)+r\prime \left(x\right)$

• Integration of Division by Polynomial: $\frac{r\left(x\right)}{q\left(x\right)}=t\left(x\right)+\frac{p\left(x\right)}{q\left(x\right)}$$\frac{r \left(x\right)}{q \left(x\right)} = t \left(x\right) + \frac{p \left(x\right)}{q \left(x\right)}$, where order of $p\left(x\right)$$p \left(x\right)$ is smaller than order of $q\left(x\right)$$q \left(x\right)$.

• Integration by Partial fractions $p\frac{x}{{\Pi }_{i=1}^{n}{g}_{i}\left(x\right)}=\sum _{i=1}^{m}\frac{1}{{h}_{i}\left(x\right)}$$p \frac{x}{{\Pi}_{i = 1}^{n} {g}_{i} \left(x\right)} = {\sum}_{i = 1}^{m} \frac{1}{{h}_{i} \left(x\right)}$ where $\frac{1}{{h}_{i}\left(x\right)}$$\frac{1}{{h}_{i} \left(x\right)}$ is integrable.

Let us restrict to this set of integrations.

This is an overview of the whole lesson.

The underlying pattern in this whole lesson is -- when we cannot work out integration of an integrand, convert the multiplication in integrand into something for which integration can be worked out. Unless you see this underlying pattern and connect the methods of integration, the results will be quite tedious and voluminous to learn.

Outline