> restart:with(student):

Integration by Change of Variable

Recall the chain rule : [f(g(x))]' = f'(g(x))g'(x). This means that F(x) = f(g(x)) is an antiderivative of an integrand of the form f'(g(x))g'(x). To integrate you can make a change of variable : Let u = g(x) and du = g'(x)dx (the differential). You then integrate with respect to the new variable u.

If you are working with a definite integral then the limits of integration are changed according to the substitution equation u = g(x).

e.g.'s

> f:=Int(4/(1+2*x)^3,x);f_new:=changevar(u=1+2*x,f1,u);value(f_new),value(f);

> f1:=Int(sec(x)*tan(x)*sqrt(1+sec(x)),x);f1_new:=changevar(u=1+sec(x),f1,u);value(f1_new),value(f1);

In the above e.g. you should also try u1 = sec(x) and u2 = to see what happens!!

> f2:=Int(cos(Pi*t),t=0..1);f2_new:=changevar(w=Pi*t,f2,w);value(f2_new),value(f2);

> f3:=Int(1/x^2*sqrt(1+1/x),x=1..4);f3_new:=changevar(q=1+1/x,f3,q);value(f3),value(f3_new);evalf(%);

e.g. You should memorize the following result:

Use substitution to find the antiderivative of tan(x):

> Int(tan(x),x)=int(tan(x),x);Int(tan(x),x)=ln(abs(sec(x)));

#75, page 417

> restart:

The rate of airflow into the lungs can be modelled by measured in L/s. Determine the volume of inhaled air at time t and the maximum lung capacity.

> f:=1*sin(2*Pi*t/5)/2;V:=int(f,t)+C;

Note: at time t the volume of inhaled air is 0 so that V(0) = 0 and we can use this to find C.

> C:=solve(subs(t=0,V)=0):V;

> plot({f,V},t=0..10,colour=[blue,red],thickness=2);

The volume curve has horizontal tangents when = 0 <=> for n = 0, 1, 2, ... It follows that max. volume will occur at 2.5 s into the cycle and the volume will be:

> evalf(subs(t=2.5,Vol_t))*`litres`;

Some Chain Rule formulae.

Power functions

> Int(u(x)^n*diff(u(x),x),x)=int(u(x)^n*diff(u(x),x),x);

e.g.

> Int(x^2/sqrt(x^3+2),x)=int(x^2/sqrt(x^3+2),x)+C;

e.g.

> Int((sin(x))^3*cos(x),x)=int((sin(x))^3*cos(x),x)+K;

Trig functions

> Int(sin(u(x))*diff(u(x),x),x)=int(sin(u(x))*diff(u(x),x),x);Int(cos(u(x))*diff(u(x),x),x)=int(cos(u(x))*diff(u(x),x),x);

e.g.

> Int(sin(x^4)*2*x^3,x)=int(sin(x^4)*2*x^3,x)+C;Int(cos(2*ln(x))/x,x)=int(cos(2*ln(x))/x,x)+K;

Exponential functions

> Int(exp(1)^(u(x))*diff(u(x),x),x)=int(exp(1)^(u(x))*diff(u(x),x),x)+C;

e.g.

> Int(x*exp(1)^(-x^2),x)=int(x*exp(1)^(-x^2),x)+C;