Contents

Definite Integrals


A simple example:

integrate (x*log(x), x, 1, 10);
					     99
(%o5) 			        50 log(10) - --
					     4

You can use symbolic integration limits, as in:

integrate (x*log(x), x, a, b);

but Maxima will ask you about the sign of the difference b-a:

To avoid this, you can specify an assumption:

assume(a > b)
integrate (x*log(x), x, a, b);
			 2	     2	    2	        2
		      2 b  log(b) - b	 2 a  log(a) - a
(%o3) 		      ---------------- - ----------------
			     4		        4

It would be interesting to evaluate the same integral under the assumption a < b. You cannot declare that assumption for as long as the assumption a > b holds. To introduce an assumption that contradicts to an earlier declared assumption, you have to remove the assumption that will be invalidated by the new assumption:

forget(a > b);
assume(a < b);
integrate (x*log(x), x, a, b);
			 2	     2	    2	        2
		      2 b  log(b) - b	 2 a  log(a) - a
(%o8) 		      ---------------- - ----------------
			     4		        4

Examples from the Moses paper:

(%i1) integrate(cos(x)^2 - sin(x), x, 0, 2*%pi);
(%o1) 							        %pi
(%i2) integrate ((x^2 + a*x + b)/(x^4 + 10*x^2 + 9), x, 0, inf);
						  4 %pi b + 12 log(3) a + 12 %pi
(%o2) 						  ------------------------------
								96
(%i3) integrate ((x^2 + a*x + b)/(x^4 + 10*x^2 + 9), x, -inf, inf);
							   %pi (2 b + 6)
(%o3) 							   -------------
								24
(%i4) integrate (sin(x)/x, x, -inf, inf);
(%o4) 							        %pi
(%i5) integrate (cos(x)/(x^2 + a^2), x, -inf, inf);
Is  a  positive, negative, or zero?

positive
;								   - a
							     %pi %e
(%o5) 							     ---------
								 a
(%i6) integrate (1/(sqrt(x)*(x + 1)), x, 0, inf);
(%o6) 							        %pi
(%i7) 


Contents