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                                    INTEGRATION BY SUBSTITUTION




Integration by substitution. A very useful device for transforming a given integrand into a form in which the fundamental integration formulas can be applied is that of substituting a new variable. Consider the following example.


Example. Find


             ole.gif


Solution. This integration cannot be performed directly by any of the standard formulas. Let us make the substitution


            x = 2 sin θ


which will, we shall see, have the effect of transforming the integrand into a rational trigonometric function.


If x = 2 sin θ then dx = 2 cos θ dθ. We then have, upon substituting,


             ole1.gif


                                     ole2.gif


                                     ole3.gif


Now using the trigonometric identity cos 2θ = 2cos2 θ - 1 we get


                                     ole4.gif


                                    = 2θ + sin 2θ + C .


We have thus performed the integration in terms of θ. We can now change the result back into terms of x as follows:


Since x = 2 sin θ,

 

             ole5.gif


Using this relationship we construct the triangle shown in Fig. 1. We know two sides of the triangle and get the other side from the Pythagorean theorem i.e. x2 + y2 = 22 so ole6.gif .


ole7.gif

The trigonometric formula for sin 2θ is


            sin 2θ = 2 sin θ cos θ .                                                  


Using Fig. 1 we have


             ole8.gif


Thus

             ole9.gif

 

 

Trigonometric substitutions. Integrands of the following types are often encountered in applications:


             ole10.gif


where m is a positive or negative integer. Integrands which are algebraic and involve no irrational function of x aside from integral powers of one of the quantities


             ole11.gif


can be transformed into rational trigonometric functions by a trigonometric substitution of the following type:


ole12.gif


ole13.gif


ole14.gif


It is easily seen that these substitutions convert the integrands to rational expressions in terms of trigonometric functions since


ole15.gif


ole16.gif


ole17.gif



ole18.gif

Appropriate triangle constructions such as those shown in Fig. 2 obviate the use of trigonometric identities in transforming integrands.


Integrands containing quantities of types 


ole19.gif


can be dealt with in a similar way. In this case make the substitutions as follows:



ole20.gif


ole21.gif


ole22.gif





Miscellaneous substitutions.  


I If an integrand is rational except for a radical of the form


             ole23.gif


it can be rationalized by a substitution as follows:



ole24.gif


ole25.gif


ole26.gif


                        substitute q + px - x2 = (α + x)2z2 or q + px - x2 = (β - x)2z2




II Any rational function of sin θ and cos θ can be replaced with a rational function of z by using the substitution


            θ = 2 tan-1 z


since


             ole27.gif



The first and second of these relationships are obtained from Fig. 3 and the third by differentiating


            θ = 2 tan-1 z . 

 

The relationships shown in Fig. 3 are obtained from the trigonometric double-angle formula



ole28.gif

ole29.gif                                                                                                  


which is equivalent to


ole30.gif


Since θ = 2 tan-1 z ,


             ole31.gif


and substituting into 2) we obtain


ole32.gif


from which we construct the triangle of Fig. 3.


 



III Theorem. If in



             ole33.gif


m, n, p, q are integers, the substitution


             ole34.gif


will rationalize


             ole35.gif


provided that


             ole36.gif


is a positive or negative integer or zero.




Evaluating integrals of types:


ole37.gif



By using the process of “completing the square”, one can write any quadratic function Ax2 + Bx + C in one of the forms (u2 + a2), (u2 - a2), (a2 - u2), (-a2 - u2), possibly multiplied by a constant, where u is some linear function of x. The integrals 1) are thus turned into standard forms.


Reduction of Ax2 + Bx + C to one of the forms (u2 + a2), (u2 - a2), (a2 - u2) or (-a2 - u2). The function Ax2 + Bx + C can be written as


             ole38.gif


where p =B/A and q = C/A. Now we complete the square on x2 + px + q:



             ole39.gif


                                     ole40.gif


Thus we have


ole41.gif



Set


             ole42.gif



and



             ole43.gif



             ole44.gif



From 2) we see that the function Ax2 + Bx + C can now be written as one of the forms (u2 + a2), (u2 - a2), (a2 - u2) or (-a2 - u2), depending on the sign of A (with the sign of A transferring).




Other substitutions.


1. The substitution ex = z reduces


             ole45.gif


2. The substitution tan x/2 = z reduces


             ole46.gif



3. The substitution x = 1/z reduces


             ole47.gif




Change of limits corresponding to a change of variable. When one does a change of variable in a definite integral he can avoid the trouble of changing the result back into terms of the original variable by making a corresponding change in the limits as illustrated in the following example.


Example. Evaluate


             ole48.gif


Solution. Let us use the substitution x = 4 sin θ. The integration is over the x-interval from x = 0 to x = 4. The corresponding interval for the new variable is found from the relation x = 4 sin θ as follows:


Putting x = 4:


            4 = 4 sin θ,      sin θ = 1,         θ = π/2



Putting x = 0:


            0 = 4 sin θ,      sin θ = 0,         θ = 0



The θ interval that corresponds to x = 0 to x = 4 is then θ = 0 to θ = π/2. Hence,


             ole49.gif


 

Substitutions in general. A considerable amount of ingenuity is required choosing a substitution that will simplify a given integral. We have covered some of the most useful transformations. However we have by no means exhausted the possibilities. One is free at any time to make any change of variable that he pleases. If the integration can be performed in terms of the new variable, the result can easily be changed back into terms of the original variable. 


Important transformations. The following list gives some transformations and their effects.



ole50.gif

 



ole51.gif

 


 

 



 




 







 




 



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