Integration Techniques

Ravinder Kumar, Aug 1, 2019

Integration by substitution, integration by parts, and integration using partial fractions are thee most basic techniques of integration. One or combination of these techniques can help integrate an integrable function when possible. Other techniques of integration in a calculus book mostly use these techniques. This book contains these three techniques of integration.

Table of Contents

  1. Integration by substitution
    1. Integration by Substitution
  2. Integration by Parts
    1. Integration by Parts
  3. Integration by Partial fractions
    1. Partial Fraction Decomposition
    2. Integration by Partial Fractions

1.

Integration by substitution

Integration is the first choice if the integrand is not directly integrable by integration rules. For substitution rule to apply integrand is typically of the form [math]g'(x) f(g(x))[/math]. The applet has a worked out example demonstrating the method.

  • 1. Integration by Substitution

1.1.

Integration by Substitution

Integration by substitution is one of the basic methods of integration. An integral of the type [br][math]\int g'\left(x\right)f\left(g\left(x\right)\right)dx[/math][br]is an example where this method may be applicable. Sometimes using substitution [math]t=g\left(x\right)[/math] may convert the integral to a form suitable for another method. Purpose of this applet is to facilitate practice of the method of integration by substitution.
– Ravinder Kumar

2.

Integration by Parts

This method is used when the substitution method is not applicable. Here, typically, integrand is product of two functions at least one of which is integrable. if the integrand is [math]\ln(x), [/math], inverse trig or inverse hyperbolic all by itself, it may be thought of as product of two functions, namely, itself and the constant function 1. Depending upon the the two factors integration by parts may have to be applied more than once.

  • 1. Integration by Parts

2.1.

Integration by Parts

Integration by parts is one of the basic methods of integration. Integration by parts, integration by substitution, and their combination provide powerful techniques of integration. Integration by parts is applicable when the integrand is a product of two functions, at least one of which is integrable and the other differentiable. There are some exceptions, for example if the integrand is [math]ln\left(x\right)[/math], it may be written as product of two functions by writing [math]ln\left(x\right)=1\times ln\left(x\right)[/math]. Similar comment applies to inverse trig functions too. Integration by parts may have to be applied more than once in some cases, for example, when an integrand is of the form [math]x^3e^x,x^2sin\left(x\right),e^xsin\left(x\right)[/math] etc. Purpose of this applet is to facilitate practice on method of integration by parts. Follow the instructions that may be viewed by checking the Instructions check box.
– Ravinder Kumar

3.

Integration by Partial fractions

This is method of choice when the integrand is a rational function (a polynomial divided by a polynomial), substitution method is not applicable and denominator can be factored into linear or/and quadratic factors.

  • 1. Partial Fraction Decomposition
  • 2. Integration by Partial Fractions

3.1.

Partial Fraction Decomposition

The purpose of this applet is to facilitate practice in finding partial fraction decomposition of a rational function, [math]\frac{f\left(x\right)}{g\left(x\right)}[/math], where degree of [math]f\left(x\right)[/math] is less than that of [math]g\left(x\right)[/math] . First, the denominator [math]g\left(x\right)[/math] is completely factored. For example, if [math]g\left(x\right)=\left(x-a\right)\left(x-b\right)\left(x-c\right)^2\left(x^2+u\right)\left(x^2+v\right)^2[/math] partial fraction decomposition of [math]\frac{f\left(x\right)}{g\left(x\right)}[/math]is of the form[br][math]\frac{f\left(x\right)}{g\left(x\right)}=\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{\left(x-c\right)}+\frac{D}{\left(x-c\right)^2}+\frac{Ex+F}{x^2+u}+\frac{Gx+H}{x^2+v}+\frac{Jx+K}{\left(x^2+v\right)^2}[/math][br]The above factorization of [math]g\left(x\right)[/math] is used to create a template of decomposition. The questions in this applet are much simpler.[br]Follow the instructions to use the applet.
– Ravinder Kumar

3.2.

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Information