Mathematics with Maple

S.Duzhin

ANSWERS TO EXERCISES OF LESSON 5

EXERCISE 1.

> int ((x-1)^3,x);

> diff(%,x);

> int(e^x/(e^x+1),x);

> diff(%,x);

> int(x*ln(x),x);

> diff(%,x);

> int(tan(x),x);

> diff(%,x);

> int(cos(x)^4,x);

> diff(%,x);

> int(x/(x^2+1)^(1/2),x);

> diff(%,x);

> f:=x^(-4)/(x^6-1)^(1/2);

> int(f,x);

> diff(%,x);

> simplify(%-f);

> int((6*x^3-19*x^2+23*x-28)/(x^4-4*x^3+3*x^2-16*x+16),x);

> int( (1+x^2)^(-2/3),x);

> int((1+x^2+x^4)^(3/2),x);

> int(sin(x)/x,x);

EXERCISE 2

> int (y/(x-y)^2,x);

> int (-x/(x-y)^2,y);

> simplify(int (y/(x-y)^2,x)-int (-x/(x-y)^2,y));

> simplify(diff (-y/(x-y),y));

EXERCISE 3

x^3+y^3/3+z^3/3-xyz

EXERCISE 4

> plot({8*x-x^2,2*x},x=-1..7);

> solve(8*x-x^2=2*x);

> int(8*x-x^2-2*x, x=0..6);

EXERCISE 5

> plot({sin(x)^2},x=-1..4);

> int (sin(x)^2, x=0..Pi);

EXERCISE 6

> plot(4/x^2,x=0.9..4);

> int (4/x^2,x=1..3);

EXERCISE 7

> plot({x^2/2, 4*sqrt(x)},x=0..5);

> solve(x^2/2=4*sqrt(x));

> int(4*sqrt(x)-x^2/2, x=0..4);

EXERCISE 8

> 2*int(sqrt(1-x^2), x=-1..1);

EXERCISE 9

Here, Maple is useful to vizualise the geometrical figures.

You see that every horizontal section by a plane z=t

is a square bounded by the lines |x|=sqrt(1-t^2) and |y|=sqrt(1-t^2). Its sides are 2*sqrt(1-t^2) and hence the area is 4*(1-t^2). Now you have to integrate this function from t=-1 to t=1. (You can do it with or without Maple, of course).

Answer: 16/3.