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mathsIntegral CalculusVarious Forms and Results of Integrals

### Integration using Identities

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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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