Westie's Workshop



Upper 6 Chapter 11IntegrationChapter Overview Integration Integrals of the form f'ax+b Using Trigonometric Identities The Reverse Chain Rule Integration by Substitution Integration by Parts Integration Using Partial Fractions Area Under a Curve The Trapezium Rule Parametric Equations Differential Equations Forming Differential Equations334010-571500003333759588400295275-15240000INTEGRATIONIntegration is the reverse of differentiation. We use known derivatives to integrate. The following are integrals that you should know:4914900294640Exercise 11A Page 29500Exercise 11A Page 295INTEGRALS OF THE FORM f'ax+bThe following are integrals that you should know:xndx=xn+1n+1+cexdx=ex+c1xdx=lnx+ccos xdx=sin x+csin xdx=-cos x+csec2xdx=tan x+ccosec x cot xdx=-cosec x+ccosec2xdx=-cot x+csec x tan xdx=sec x+cUSING TRIGONOMETRIC IDENTITIESThe following are identities that you should know:sinA±B=cosA±B=tanA±B=sin 2A=cos 2A=cos 2A=cos 2A=tan 2A=sec2A=cosec2A=We can use these identities to transform an expression that cannot be integrated into one that can be integrated.These first examples focus on manipulation of the identities rather than integration.Examples sin4x=2 sin3xcos3x=cos5x=4 cos23x-2=Further examples Show thatπ12π8sin2xdx=π48+1-284543425255905Exercise 11C Page 30000Exercise 11C Page 300 Ex 11D Page 302-303More examplesUse the substitution u2=x+1 to find x x + 112dxExample 4Use the substitution x=23tanu to find 1 4+9x2dxEdexcel will usually give you the substitution in the exam question.However, if you are not provided with a substitution, a ‘rule of thumb’ is to replace expressions inside roots, powers or the denominator of a fraction by the variable u.INTEGRATION BY SUBSTITUTION AND DEFINITE INTEGRALSWhen you use integration by substitution to evaluate a definite integral, you do not need to rewrite the expression in terms of x. However, if you use the expression in terms of u, you must replace the x limits with u limits.Alternatively, you could convert the integral back to a function of x and use the original limits but this is usually messier!Example 5Calculate 0π2cosx1+sinx?dx-342900274955Example 631813502101854819650112395Exercise 11E Page 30600Exercise 11E Page 306Example 1Example 2Findxln x dxHere, the choice of u must be lnx because lnx is difficult to integrateExample 3Findln x dxHere, the ‘trick’ is to write the integral as 1 ×ln x dxAgain, the choice of u must be lnxExample 5Findexcos x dxleft240191000 ................
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