Maple Tutor

Part 10: Integration

1. First we calculate indefinite integrals. If necessary, unassign x:
   x:='x'
   Then enter
   x*sin(x)
   Use your mouse to highlight this expression, click the right mouse button, and select "Integrate"  > "x" from the popup menus.  Maple should make an arrow pointing to the desired output. (In Maple versions 11 and higher the arrow has a label.)
   
   Use the "Differentiate" option in the popup menu to check that the result is a correct antiderivative of x sin(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 to find an antiderivative of sin(x²).  The Fresnel function is known to Maple, but probably not to you. However, you can check by differentiation that the result is a correct antiderivative.

3. Next we calculate definite integrals. To integrate x sin(x) over the interval [0,π/2], we will use the palettes again.  Click on the palette button which says "Expression", and select the definite integral symbol from the grid of symbols which appears.
   Maple will insert a definite integral template into the document with places to enter the limits of integration, the integrand, and the variable of integration.
   
   
   1. The cursor is positioned for you to enter the lower limit of integration. Enter 0.
      
      
   2. Press the Tab key. Notice that the cursor moves to the upper limit of integration. In general, the Tab key cycles you through the places in the template where entries are needed. Now, enter π/2, then Tab, then x sin(x), then Tab again, and finally x.
      
      
   3. Now, if you press Enter, Maple will calculate the value of the integral.

4. Now try this method on the integral of sin(x³+x⁵) over the interval [0,π/2]. Maple still doesn't know an antiderivative for sin(x³+x⁵). To obtain a numerical estimate, use the "Approximate" option in the popup menu.

   If you know that all you want is a numerical estimate, you do not have to press Enter before you use the "Approximate" option.  This may save you some time because 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. (The infinity symbol is in the "Common Symbols" palette.)
   • 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.