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On looking at the integral, we see that the expression cos x is the derivative of sin x This observation suggests the substitution sin x = u Thus cos x dx = du We must now substitute these expressions into the integral, replacing all x-expressions with u-expressions When we are through with this process, no x expressions can remain The result is u5 du This is of course an easy integral for us So we have [sin x]5 cos x dx = u5 du = u6 + C 6

The output is shown here:

Here, Derived3 inherits Derived2, which inherits Derived1, which inherits Base As the output verifies, since Who( ) is not overridden by either Derived3 or Derived2, it is the override of Who( ) in Derived1 that is executed, since it is the first version of Who( ) that is found One other point: Properties can also be modified by the virtual keyword and overridden using override The same is true for indexers

.

As mentioned at the start of this section, the generic collection classes largely parallel their non-generic relatives, although in some cases the names have been changed Also, some differences in organization and functionality exist The generic collections are defined in

Now the important final step is to resubstitute the x-expressions in place of the u-expressions The result is then [sin x]5 cos x dx = sin6 x + C 6

25:

SystemCollectionsGeneric The ones described in this chapter are shown in Table 25-14 These classes form the core of the generic collections

NOTE SystemCollectionsGeneric also includes the following classes:

SynchronizedCollection<T> is a synchronized collection based on IList<T> SynchronizedReadOnlyCollection<T> is a read-only synchronized collection based on IList<T> SynchronizedKeyedCollection<K, T> is an abstract class used as a base class by SystemServiceModelUriSchemeKeyedCollection KeyedByTypeCollection<T> is a collection that uses types as keys

MATH NOTE Always be sure to check your work You can differentiate the answer in the last example to recover the integrand, confirming that the integration has been performed correctly EXAMPLE

The List<T> class implements a generic dynamic array and is conceptually similar to the non-generic ArrayList class List<T> implements the ICollection, ICollection<T>, IList, IList<T>, IEnumerable, and IEnumerable<T> interfaces List<T> has the constructors shown here: public List( ) public List(IEnumerable<T> collection) public List(int capacity) The first constructor builds an empty List with a default initial capacity The second constructor builds a List that is initialized with the elements of the collection specified by collection and with an initial capacity at least equal to the number of elements The third constructor builds an array list that has the specified initial capacity The capacity is the size of the underlying array that is used to store the elements The capacity grows automatically as elements are added to a List<T> Each time the list must be enlarged, its capacity is increased

Class Dictionary<TKey, TValue> HashSet<T> LinkedList<T> List<T> Queue<T> SortedDictionary<TKey, TValue> SortedList<TKey, TValue> SortedSet<T> Stack<T> Description Stores key/value pairs Provides functionality similar to that found in the non-generic Hashtable class Stores a set of unique values using a hash table Stores elements in a doubly linked list A dynamic array Provides functionality similar to that found in the non-generic ArrayList class A first-in, first-out list Provides functionality similar to that found in the non-generic Queue class A sorted list of key/value pairs A sorted list of key/value pairs Provides functionality similar to that found in the non-generic SortedList class A sorted set A first-in, last-out list Provides functionality similar to that found in the non-generic Stack class

TABLE 25-14