Maple Tutor

Part 10: Integration

1. First we calculate indefinite integrals. If necessary, unassign x:
   x:='x';
   Then enter
   int(x*sin(x),x);
   Check that the result is an antiderivative of x sin(x):

   Now enter
   diff(%,x);

2. Now try to find an antiderivative for sin(x³ + x⁵). Maple does not know an antiderivative of this function that may be defined in terms of functions known to it. On the other hand, try
   int(sin(x^2),x);
   The Fresnel function is known to Maple, but probably not to you. However, you can check by differentiation that it is an antiderivative.

3. Next we calculate definite integrals. To integrate x sin(x) over the interval [0,pi/2], enter
   int(x*sin(x),x=0..Pi/2);

4. Now try this method on the integral of sin(x³+x⁵) over the interval [0,pi/2]. Maple still doesn't know an antiderivative for sin(x³+x⁵). To obtain a numerical estimate, enter
   evalf(int(sin(x^3+x^5),x=0..Pi/2));

   If you know that all you want is a numerical estimate, you can enter
   evalf(Int(sin(x^3+x^5),x=0..Pi/2));
   The significance of the upper-case I in Int is that Maple does not try to find a symbolic solution before starting on the numerical estimate.

5. Use Maple to find the exact value of each of the following integrals. (Write out "infinity" for the infinity symbol.)

• The integral of 1/(1+x²) from 0 to 1,

• The integral of 1/(1+x²) from 0 to infinity,

• The integral of 1/(1+x⁴) from 0 to infinity.

Note: In Release 5 of Maple V, you have access to three palettes -- Symbol Palette, Expression Palette, and Matrix Palette. These palettes simplify the creation of many common Maple commands. The Expression Palette is particularly useful for setting up integrals. The use of these palettes is described in the Appendix.