Math 174

Calculus II

Fall 2002

Dr. Constant J. Goutziers

Department of Mathematics, Computer Science and Statistics

goutzicj@oneonta.edu

Lesson 13

Rationalizing Substitutions

Initializations

>    restart;
with(student):
with(oneonta):

>   

14.1  Rationalizing Substitutions involving Radicals and Trigonometric Functions.

Examples

Example 13.1.1
Evaluate Int(1/(x^(1/2)+x^(1/3)),x)

>    e1:=Int(1/(x^(1/2)+x^(1/3)), x);

e1 := Int(1/(x^(1/2)+x^(1/3)),x)

>    e2:=changevar(x=u^6, e1, u);

e2 := Int(6/((u^6)^(1/2)+(u^6)^(1/3))*u^5,u)

>    e3:=simplify(e2, symbolic);

e3 := 6*Int(u^3/(u+1),u)

>    e4:=parf(e3);

e4 := 6*Int(u^2-u+1-1/(u+1),u)

>    e5:=value(e4)+c;

e5 := 2*u^3-3*u^2+6*u-6*ln(u+1)+c

>    e6:=subs(u=x^(1/6), e5);

e6 := 2*x^(1/2)-3*x^(1/3)+6*x^(1/6)-6*ln(x^(1/6)+1)+c

>   

Example 13.1.2
Evaluate Int(1/(cos(x)+2*sin(x)),x) .

>    e1:=Int(1/(cos(x)+2*sin(x)), x);

e1 := Int(1/(cos(x)+2*sin(x)),x)

>    e2:=changevar(u=tan(x/2), e1, u);

e2 := Int(2/(cos(2*arctan(u))+2*sin(2*arctan(u)))/(1+u^2),u)

>    e3:=expand(e2);

e3 := 2*Int(1/((2/(1+u^2)-1+4*u/(1+u^2))*(1+u^2)),u)

>    e4:=factor(e3);

e4 := 2*Int(-1/(-1+u^2-4*u),u)

>    e5:=completesquare(e4, u);

e5 := 2*Int(-1/((u-2)^2-5),u)

Even though this denominator can be factored, and the integral treated with partial fraction decomposition,

>    e51:=2*Int(factor(op(1, op(2, e5)), 5^(1/2)), u);

e51 := 2*Int(-1/((u-2-5^(1/2))*(u-2+5^(1/2))),u)

>    e52:=parf(e51);

e52 := 2*Int(1/10*5^(1/2)/(u-2+5^(1/2))-1/10*5^(1/2)/(u-2-5^(1/2)),u)

we will simply ask for its value.

>    e6:=value(e5)+c;

e6 := 2/5*5^(1/2)*arctanh(1/10*(2*u-4)*5^(1/2))+c

>    e7:=convert(e6, ln);

e7 := 2/5*5^(1/2)*(1/2*ln(1/5*(u-2)*5^(1/2)+1)-1/2*ln(1-1/5*(u-2)*5^(1/2)))+c

>    e8:=simplify(e7);

e8 := 1/5*5^(1/2)*ln(u-2+5^(1/2))-1/5*5^(1/2)*ln(-u+2+5^(1/2))+c

>    e9:=subs(u=tan(x/2), e8);

e9 := 1/5*5^(1/2)*ln(tan(1/2*x)-2+5^(1/2))-1/5*5^(1/2)*ln(-tan(1/2*x)+2+5^(1/2))+c

>