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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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In this page, integration using identities is explained with examples.



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Consider the integration `int (x+2)(x+3) dx`. Which of the following would help in working out the integral?

  • use an identity to convert the multiplication to a sum of terms
  • use an identity to convert the multiplication to a sum of terms
  • a polynomial in factors form cannot be integrated

The answer is "use an identity to convert the multiplication to a sum of terms"

`int (x+2)(x+3) dx`.

use the identity `(x+a)(x+b) ``= x^2+(a+b)x+ab`
`=int [x^2 + (2+3)x + 2xx3 ]dx`

This integral can be computed for each of the terms.

Consider the integration `int cos^2 x dx`. Which of the following would help in working out the integral?

  • use a trigonometric identity to convert `cos^2 x` into sum of terms
  • use a trigonometric identity to convert `cos^2 x` into sum of terms
  • only numerical integration is possible

The answer is "use a trigonometric identity to convert `cos^2 x` into sum of terms "

We know the trigonometric identity `cos2x = cos^2 x - sin^2 x`

substitute `sin^2 x = 1-cos^2x` and rearrange the terms
`cos^2 x = (1+cos2x)/2`

`int cos^2 x dx`

`=int [1/2 + (cos2x)/2 ]dx `

This integral can be computed for each of the terms.

Some trigonometric identities useful for integration are

`2sinxcosy ``= sin(x+y)+sin(x-y)`
`2sinxcosx ``= sin(x+y)`

`2cosxcosy ``= cos(x-y)+cos(x+y)`
`2cos^2x ``= 1+cos(2x)`
`4cos^3x ``= 3cosx + cos3x`

`2sinxsiny ``= cos(x-y)-cos(x+y)`
`2sin^2x ``= 1-cos(2x)`
`sin^3x ``= 3sinx - sin3x`

When the integrand is product of several functions, convert that to sum of functions using known identities.

Integration using Identities: when `f(x) = Pi_(i=1)^n g_i(x)` Use identities to convert to `f(x) = sum_(i=1)^m h_i(x)`, such that integration can be individually performed on `h_i(x)`.

Solved Exercise Problem:

Integrate `int cos^2 x dx + int sin^2 x dx`.

  • use the identity `cos^2x + sin^2 x = 1`
  • use the identity `cos^2x + sin^2 x = 1`
  • cannot use the identity `cos^2x + sin^2 x = 1`

The answer is "use the identity `cos^2x + sin^2 x = 1`"

                            
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