__Indefinite and Definite Integrals__

We want to find the indefinite integral of one of the usual textbook examples, e.g.:

`> `
**f := x^2*sin(x)*exp(-3*x);**

`> `
**int(f, x);**

To test the result, we differentiate it:

`> `
**diff(%,x);**

`> `
**simplify(%);**

Next we try a rational function:

`> `
**g := (2*x^2-3*x+1)/(x^5-7*x^4+17*x^3-19*x^2+16*x-12);**

`> `
**int(g, x);**

To see what Maple did, we look at the partial fraction decomposition:

`> `
**factor(g);**

`> `
**convert(g, parfrac, x);**

Finally we want to compute the circumference of an ellipse of excentricity eps:

The parameter form of the ellipse leads to the following integral for the circumference:

`> `
**f1 := sqrt(1+eps^2*sin(t)^2);**

`> `
**l_ell := 4*int(f1, t=0..Pi/2);**

The integral can not be solved with the usual functions. Since it appears in many different contexts, one has given a special name to it: "complete elliptic integral of the second kind". We look at its taylor expansion and function graph:

`> `
**taylor(%, eps=0, 5);**

For the plot of the approximation we have to throw away the O(..) term:

`> `
**l_approx := convert(%, polynom);**

`> `
**plot({l_ell,l_approx}, eps=-2..2);**