Integral Calculus : Integration using Identities

maths > integral-calculus

Integration using Identities

what you'll learn...

Overview

Integration using Identities :

when $f (x) = Π_{i = 1}^{n} g_{i} (x)$ $f(x) = Pi_(i=1)^n g_i(x)$ , use identities to convert to $f (x) = \sum_{i = 1}^{m} h_{i} (x)$ $f(x) = sum_(i=1)^m h_i(x)$ , such that integration can be individually performed on $h_{i} (x)$ $h_i(x)$

identity

Consider the integration $\int (x + 2) (x + 3) d x$ $int (x+2)(x+3) dx$ . To work out the integral use an algebraic identity to convert the multiplication to a sum of terms.

$\int (x + 2) (x + 3) d x$ $int (x+2)(x+3) dx$ .

use the identity $(x + a) (x + b)$ $(x+a)(x+b)$ $= x^{2} + (a + b) x + a b$ $= x^2+(a+b)x+ab$
$= \int [x^{2} + (2 + 3) x + 2 \times 3] d x$ $=int [x^2 + (2+3)x + 2xx3 ]dx$

This integral can be computed for each of the terms.

cos squared

Consider the integration $\int \cos^{2} x d x$ $int cos^2 x dx$ .

We know the trigonometric identity $\cos 2 x = \cos^{2} x - \sin^{2} x$ $cos2x = cos^2 x - sin^2 x$

substitute $\sin^{2} x = 1 - \cos^{2} x$ $sin^2 x = 1-cos^2x$ and rearrange the terms
$\cos^{2} x = \frac{1 + \cos 2 x}{2}$ $cos^2 x = (1+cos2x)/2$

$\int \cos^{2} x d x$ $int cos^2 x dx$

$= \int [\frac{1}{2} + \frac{\cos 2 x}{2}] d x$ $=int [1/2 + (cos2x)/2 ]dx$

This integral can be computed for each of the terms.

some identities

Some trigonometric identities useful for integration are

$2 \sin x \cos y$ $2sinxcosy$ $= \sin (x + y) + \sin (x - y)$ $= sin(x+y)+sin(x-y)$
$2 \sin x \cos x$ $2sinxcosx$ $= \sin (x + y)$ $= sin(x+y)$

$2 \cos x \cos y$ $2cosxcosy$ $= \cos (x - y) + \cos (x + y)$ $= cos(x-y)+cos(x+y)$
$2 \cos^{2} x$ $2cos^2x$ $= 1 + \cos (2 x)$ $= 1+cos(2x)$
$4 \cos^{3} x$ $4cos^3x$ $= 3 \cos x + \cos 3 x$ $= 3cosx + cos3x$

$2 \sin x \sin y$ $2sinxsiny$ $= \cos (x - y) - \cos (x + y)$ $= cos(x-y)-cos(x+y)$
$2 \sin^{2} x$ $2sin^2x$ $= 1 - \cos (2 x)$ $= 1-cos(2x)$
$\sin^{3} x$ $sin^3x$ $= 3 \sin x - \sin 3 x$ $= 3sinx - sin3x$

example

Integrate $\int \cos^{2} x d x + \int \sin^{2} x d x$ $int cos^2 x dx + int sin^2 x dx$ .

This can be integrated using the identity $\cos^{2} x + \sin^{2} x = 1$ $cos^2x + sin^2 x = 1$

summary

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

Outline

The outline of material to learn "Integral Calculus" is as follows.

• Detailed outline of Integral Calculus

→ Application Scenario

→ Integration First Principles

→ Graphical Meaning of Integration

→ Definition of Integrals

→ Fundamental Theorem of Calculus

→ Algebra of Integrals

→ Antiderivatives: Standard results

→ Integration of Expressions

→ Integration by Substitution

→ Integration using Identities

→ Integration by Parts

→ Integration by Partial Fraction

→ Integration: Combination of Methods