Courses in Math - Indefinite Integrals

Course Content

Mathematical principles of indefinite integrals are presented. This course starts with the definition of integral and its main properties. Rules of both integration by substitution and integration by parts are shown. Study of given examples can help to train analytic techniques of integration.

1. Definition
2. Main Properties
3. Table of Elementary Integrals
4. The Rules of Integration
5. Elementary Examples
6. Integration By Substitution
   • Introduction
   • Examples 1-10
   • Examples 11-20
   • Examples 21-30
   • Examples 31-40
7. Integration of Rational Functions
   • Main Types
   • Fraction Expansion – Denominators
   • Fraction Expansion – Coefficients
   • Examples 1-10
   • Examples 11-20
   • Examples 21-30
   • Ostrogradsky's Formula
8. Integration of Rational Functions with Roots
   • Rational functions of R(x, [(ax + b)/(cx + d)]^(1/m)) type
   • Functions of x^m(a + bxⁿ)^p type
   • Examples 1-10
   • Rational functions of R(x, (ax² + bx + c)^(1/2)) type. Euler substitution.
   • Examples 1-10
   • Examples 11-20
9. Integration of Trigonometric Functions
   • Functions of R(sin(x), cos(x)) and R(sin^ν(x), cos^μ(x)) types.
   • Examples 1-10
   • Examples 11-20
   • Examples 21-30
   • Examples 31-40
10. Integration By Parts
    • Introduction
    • Examples 1-10
    • Examples 11-20
    • Examples 21-30
11. Literature.

    G. M. Fichtenholz – Integral and differential calculus, vol. 2, PWN Warsaw 1999 (originally in Russian)

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How does the course work?

The course is divided into paragraphs. The first part of tutorial presents the elementary idea of finding indefinite integrals together with basic examples (as shown below). The two main ways of integration i. e. integration by substitution and integration by parts are considered. Each main part of handbook follows brief theoretical introduction (e. g. Indefinite integrals. Definition.). This course explains simple and more complicated cases of integration. Among the advanced cases are: integration of rational functions, integration of functions with roots and integration of trigonometric functions. In these examples transformations must be applied to starting expressions to obtain better form of integrated functions that, in turn, can be easily solved.
Examples presented-below show the way of putting forward solutions. Each essential part of finding the output is treated separately. Thus one can make use of only part of hints and after finding the solution compare this result with the outcome presented in the tutorial.

1. Direct Integration

   Let's consider a simple integral of a trigonometric function.

   ∫

   tg2(x)dx

   As the first step we transform the trigonometric function to a more convenient form:

   tg²(x)dx = sin²(x)/cos²(x)dx = [1 - cos²(x)]/cos²(x) = 1/cos²(x) - 1

   After that finding the solution is trivial:

   ∫

   dx/cos2(x) -

   ∫

   dx = tg(x) - x + C

2. Integration By Parts

   The second example is an example of integration by parts.

   ∫	xsin2(x)dx

   by parts transformation:

   f(x) = x; g'(x) = sin²(x)
   f'(x) = dx; g(x) = 1/2 x - 1/4 sin(2x)

   and the solution reads:

   x2/2 - x/4 sin(2x) - 1/2

   ∫

   xdx + 1/4

   ∫

   sin(2x)dx =

   x2/2 - x/4 sin(2x) - x2/4 - 1/8 cos(2x) + C =

   x2/4 - x/4 sin(2x) - 1/8 cos(2x) + C

3. Integration By Substitution

   Another example shows the main idea of integration by substitution.

   ∫	xdx/[1 - x2]1/2

   the obvious way of finding solution is to make the substitution:

   t = 1 - x²; dt = -2xdx

   the solution is:

   -1/2

   ∫

   dt/t1/2 = -t1/2 + C = -(1 - x2)1/2 + C

4. Integration by Substitution.

   Integration by substitution - second example, trigonometric function.

   ∫	sin(1/x)dx/x2

   substitution:

   t = 1/x; dt = -dx/x²

   the solution is:

   ∫	-sin(t)dt = cos(t) + C = cos(1/x) + C