review-solutions.mw

> restart;
with(Student:-Calculus1):
infolevel[Student[Calculus1]] := 1:
Understand(Int, constant, constantmultiple, power, sum, sin, cos);
Clear(all);
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> </COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> hint :=  Hint(%);
Rule[hint](GetProblem());
while  hint <>[] do
hint :=  Hint(GetProblem());
Rule[hint](GetProblem());
end do;
 

Creating problem #1
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution x = u^3, u = x^(1/3) with dx = 3*u^2*du, du = 1/3/x^(2/3)*dx 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution u = u1-1 with du=du1 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u1 = 1+u 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u = x^(1/3) 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

This problem is complete 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> Int((tan(x))^3,x);
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> hint := Hint(%);
Rule[hint](GetProblem());
while  hint <>[] do
hint :=  Hint(GetProblem());
Rule[hint](GetProblem());
end do;
 

Creating problem #2
 

Reduce the power on tan(x) using the identity tan(x)^2 = sec(x)^2-1. 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Since the derivative of sec(x) is sec(x)*tan(x), we can use the substitution u = sec(x). 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution x = arcsec(u), u = sec(x) with dx = 1/u^2/(1-1/u^2)^(1/2)*du, du = sec(x)*tan(x)*dx 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u = sec(x) 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Since the derivative of cos(x) is -sin(x), we can use the substitution u = cos(x). 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution x = arccos(u), u = cos(x) with dx = -1/(1-u^2)^(1/2)*du, du = -sin(x)*dx 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u = cos(x) 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

This problem is complete 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> </COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

>
hint :=  Hint(%);
Rule[hint](GetProblem());
while  hint <>[] do
hint :=  Hint(GetProblem());
Rule[hint](GetProblem());
end do;
 

Creating problem #3
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution x = -1/2*u, u = -2*x with dx = -1/2*du, du = -2*dx 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

This is one of the basic integrals. 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u = -2*x 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

This problem is complete 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> Int(x^2 * ln(x), x);
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> hint :=  Hint(%);
Rule[hint](GetProblem());
while  hint <>[] do
hint :=  Hint(GetProblem());
Rule[hint](GetProblem());
end do;
 

Creating problem #4
 

Integrals of the form Int(a(x)*g(b(x))^k, x), where g is a logarithmic function, inverse trigonometric function, or inverse hyperbolic function, can often be simplified using integration by parts with u = g(b(x))^k. 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

This problem is complete 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> Int(sin(x)^3 *cos(x)^4, x);
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> hint :=  Hint(%);
Rule[hint](GetProblem());
while  hint <>[] do
hint :=  Hint(GetProblem());
Rule[hint](GetProblem());
end do;
 

Creating problem #5
 

Split off an even power of sin(x) and use the identity sin(x)^2 = 1-cos(x)^2. 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Since the derivative of cos(x) is -sin(x), we can use the substitution u = cos(x). 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution x = arccos(u), u = cos(x) with dx = -1/(1-u^2)^(1/2)*du, du = -sin(x)*dx 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u = cos(x) 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Since the derivative of cos(x) is -sin(x), we can use the substitution u = cos(x). 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution x = arccos(u), u = cos(x) with dx = -1/(1-u^2)^(1/2)*du, du = -sin(x)*dx 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u = cos(x) 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

This problem is complete 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> </COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> hint :=  Hint(%);
Rule[hint](GetProblem());
while  hint <>[] do
hint :=  Hint(GetProblem());
Rule[hint](GetProblem());
end do;
 

Creating problem #6
 

Hints:

1. Integrals involving expressions of the form (x^2+a^2)^n or sqrt(x^2+a^2)^n can often be simplified using the substitution x = a*tan(u).

2. Use the substitution 4+x^2 = u^2.

3. Integrals involving expressions of the form sqrt(x^2+a^2) can often be simplified using the substitution u = sqrt(x^2+a^2)-x.
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution x = 2*tan(u), u = arctan(1/2*x) with dx = (2+2*tan(u)^2)*du, du = 1/2/(1/4*x^2+1)*dx 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Split off an even power of tan(u) and use the identity tan(u)^2 = sec(u)^2-1. 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Since the derivative of sec(u) is sec(u)*tan(u), we can use the substitution u1 = sec(u). 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution u = arcsec(u1), u1 = sec(u) with du = 1/u1^2/(1-1/u1^2)^(1/2)*du1, du1 = sec(u)*tan(u)*du 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u1 = sec(u) 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Since the derivative of sec(u) is sec(u)*tan(u), we can use the substitution u1 = sec(u). 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution u = arcsec(u1), u1 = sec(u) with du = 1/u1^2/(1-1/u1^2)^(1/2)*du1, du1 = sec(u)*tan(u)*du 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u1 = sec(u) 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u = arctan(1/2*x) 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

This problem is complete 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> </COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> hint :=  Hint(%);
Rule[hint](GetProblem());
while  hint <>[] do
hint :=  Hint(GetProblem());
Rule[hint](GetProblem());
end do;
 

Creating problem #7
 

Hints:

1. Integrals involving expressions of the form (x^2+a^2)^n or sqrt(x^2+a^2)^n can often be simplified using the substitution x = a*tan(u).

2. Use the substitution 4+x^2 = u^2.

3. Integrals involving expressions of the form sqrt(x^2+a^2) can often be simplified using the substitution u = sqrt(x^2+a^2)-x.
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution x = 2*tan(u), u = arctan(1/2*x) with dx = (2+2*tan(u)^2)*du, du = 1/2/(1/4*x^2+1)*dx 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Make change of variable. 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution u = arccos(u1), u1 = cos(u) with du = -1/(1-u1^2)^(1/2)*du1, du1 = -sin(u)*du 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u1 = cos(u) 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u = arctan(1/2*x) 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

This problem is complete 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> </COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

> hint :=  Hint(%);
Rule[hint](GetProblem());
while  hint <>[] do
hint :=  Hint(GetProblem());
Rule[hint](GetProblem());
end do;
 

Creating problem #8
 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution x = 1+u with dx=du 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u = x-1 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution x = 1+u with dx=du 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u = x-1 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Rewrite the numerator in a form which contains the derivative of the denominator 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Note that the derivative of 4+x^2 is 2*x, so we can make a change of variable. 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution x = (u-4)^(1/2), u = 4+x^2 with dx = 1/2/(u-4)^(1/2)*du, du = 2*x*dx 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u = 4+x^2 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Integrals involving expressions of the form (x^2+a^2)^n or sqrt(x^2+a^2)^n can often be simplified using the substitution x = a*tan(u). 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Applying substitution x = 2*tan(u), u = arctan(1/2*x) with dx = (2+2*tan(u)^2)*du, du = 1/2/(1/4*x^2+1)*dx 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

Reverting substitution using u = arctan(1/2*x) 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

This problem is complete 

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

</COMMENT> No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!  

>