# 9.1  Integration by Substitution (Change of Variables)
# Art Belmonte, Summer 1996
# These are the shaded gray examples which appear in Section 9.1.
# In-line plots have been removed to conserve disk space.
> restart; with(student):
> A:=Int(1 / sqrt(4 + 9*x^2), x);

                             /
                            |        1
                      A :=  |  ------------- dx
                            |          2 1/2
                           /   (4 + 9 x )

> changevar(3*x=2*tan(theta), A, theta); Atemp:=value(");

                  /                       2
                 |        2 + 2 tan(theta)
                 |  1/3 ---------------------- dtheta
                 |                       2 1/2
                /       (4 + 4 tan(theta) )


                   Atemp := 1/3 arcsinh(tan(theta))

# Here we alternatively use changevar again, which eliminates the need
# for simplify (in this case).
> Avalue:=changevar(theta=arctan(3*x/2), Atemp, x);

                     Avalue := 1/3 arcsinh(3/2 x)

# Again, simplify was not needed (in this case, with Maple V Release 4).
> diff(Avalue, x);

                                  1
                            -------------
                                    2 1/2
                            (4 + 9 x )

> changevar(3*x=2*tan(theta), Int(1 / sqrt(4 + 9*x^2), x), theta);

                  /                       2
                 |        2 + 2 tan(theta)
                 |  1/3 ---------------------- dtheta
                 |                       2 1/2
                /       (4 + 4 tan(theta) )

> Int(1 / sqrt(4 + 9*x^2), x); value(");

                           /
                          |        1
                          |  ------------- dx
                          |          2 1/2
                         /   (4 + 9 x )


                          1/3 arcsinh(3/2 x)

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