Differentiation and integration

>> syms x y >> y = atan(x)Differentiating

>> diff(y,x) ans = 1/(1+x^2) >> diff(y,x,3) ans = 8/(1+x^2)^3*x^2-2/(1+x^2)^2The indefinite integral

>> int(y,x) ans = x*atan(x)-1/2*log(x^2+1)The definite integral

>> int(y,x,2,7) ans = 7*atan(7)-1/2*log(2)-1/2*log(5)-2*atan(2)This is the exact value of the integral. The numerical value is obtained by

>> double(ans) ans = 6.6367The command

`double`

gives the numerical value of an exact number.
By far most integrals cannot be calculated explicitly. Consider the example of trying to calculate the integral of `

>> int(exp(-x)*sqrt(1+x^3),x,1,4) ans = int(exp(-x)*(1+x^3)^(1/2),x = 1..4) >> double(ans) ans = 1.0017MATLAB returns the integral, because it cannot calculate it explicitly. The function

`double`

gives a numerical approximation of the integral.

Esteur 2010-03-22