% Problem 10

% Clear variables and declare x to be symbolic.

clear
syms x

% a)

int(sin(x), 0, pi/2)
 
ans =
 
1
 
% The integral is 1, which we can easily check by hand.

% b)

int(x*cos(x^2))
 
ans =
 
1/2*sin(x^2)
 
% That looks correct, let's check by differentiating the answer.

diff(ans)
 
ans =
 
x*cos(x^2)
 
% Bingo!

% c)

int(sin(3*x)*sqrt(1 - cos(3*x)))
 
ans =
 
2/9*(1-cos(3*x))^(3/2)
 
diff(ans)
 
ans =
 
sin(3*x)*(1-cos(3*x))^(1/2)
 
% Right again.

% d)

int(log(x))
 
ans =
 
x*log(x)-x
 
diff(ans)
 
ans =
 
log(x)
 
% And again.

% e)

int(x^2*sqrt(x + 4))
 
ans =
 
2/7*(x+4)^(7/2)-16/5*(x+4)^(5/2)+32/3*(x+4)^(3/2)
 
diff(ans)
 
ans =
 
(x+4)^(5/2)-8*(x+4)^(3/2)+16*(x+4)^(1/2)
 
% This time it's not so clear that the derivative of the integral
% is equivalent to the original function.  Let's simplify the answer
% to make sure.

simplify(ans)
 
ans =
 
x^2*(x+4)^(1/2)
 
% That looks better.

% f)

int(sqrt(x^4 + 1))
 
ans =
 
1/3*x*(x^4+1)^(1/2)+int(2/3/(x^4+1)^(1/2),x)
 
% This is rather difficult to read; let's make it easier with pretty.

pretty(ans)
 
                                        /
                          4     1/2    |           1
                  1/3 x (x  + 1)    +  |  2/3 ----------- dx
                                       |        4     1/2
                                      /       (x  + 1)

% It looks like MATLAB integrated by parts and performed some other
% manipulations, then gave up; the answer is given in terms of another
% indefinite integral.  Nonetheless, we can differentiate this
% expression.

diff(ans)
 
ans =
 
1/3*(x^4+1)^(1/2)+2/3*x^4/(x^4+1)^(1/2)+2/3/(x^4+1)^(1/2)
 
% Again we have to simplify the answer.

simplify(ans)
 
ans =
 
(x^4+1)^(1/2)
 
% Correct!

% g)

int(exp(cos(x)))

Warning: Explicit integral could not be found.
> In /usr/local/matlab5/toolbox/symbolic/@sym/int.m at line 58
  In /www/users/rll/courses/math246.f98/psa/probA10.m at line 67
 
ans =
 
int(exp(cos(x)),x)
 
% This is another integral that MATLAB can't perform symbolically.

diff(ans)
 
ans =
 
exp(cos(x))
 
% Not suprising.

% h)

int(exp(-x^2), -inf, inf)
 
ans =
 
pi^(1/2)
 
% The exact value of this definite integral is the square root of pi.

echo off