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Various Forms and Results of Integrals

Various Forms and Results of Integrals

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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)`.

Integration using Identities

plain and simple summary

nub

plain and simple summary

nub

dummy

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

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simple steps to build the foundation

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


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Starting on learning "". In this page, integration using identities is explained with examples.

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
  • 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
  • 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`

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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)`.



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

  • 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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Consider the integration, integral x + 2 multiplied x + 3 d x. Which of the following would help in working out the integral.
use;identity;convert;multiplication
use an identity to convert the multiplication to a sum of terms
polynomial;factors;form;not;
a polynomial in factors form cannot be integrated
The answer is "use an identity to convert the multiplication to a sum of terms"
Integration is worked out as given.
Consider the integration: integral cos squared x dx. Which of the following would help in working out the integral.
use;trigonometric;identity;convert
use a trigonometric identity to convert cos squared x into sum of terms
only;numerical;possible
only numerical integration is possible
The answer is "use a trigonometric identity to convert cos squared x into sum of terms"
We know the trigonometric identity cos 2 x = cos squared x minus sine squared x. The integration is worked out using that.
Some trigonometric identities useful for integration are given.
Simplified nub: When the integrand is product of several functions, convert that to sum of functions using known identities.
Reference Jogger: Integration using Identities is summarized.
Integrate the given expression.
1
2
The answer is "use the identity cos squared x + sin squared x = 1"

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