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SOLUTION 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

Property public IComparer<TK> Comparer { get; } public SortedDictionary<TK, TV>KeyCollection Keys { get; } public SortedDictionary<TK, TV>ValueCollection Values { get; } Description Obtains the comparer for the invoking dictionary Obtains a collection of the keys

I yxi = -

Notice that the keys and values contained within the collection are available as separate lists through the Keys and Values properties The types SortedDictionary<TK, TV>KeyCollection SortedDictionary<TK, TV>ValueCollection are collections that implement both the generic and non-generic forms of ICollection and IEnumerable SortedDictionary<TK, TV> defines the following indexer (which is specified by IDictionary<TK, TV>): public TV this[TK key] { get; set; } You can use this indexer to get or set the value of an element You can also use it to add a new element to the collection Notice that the index is not actually an index, but rather the key of the item

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From Eqs (754a to 756) it is noted that all components of the inertia matrix are given as linear combinations of four integrals, namely, Ii(1)

Now the important nal 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

When enumerated, SortedDictionary<TK, TV> returns key/value pairs in the form of a KeyValuePair<TK, TV> structure Recall that this structure defines the following two fields: public TK Key; public TV Value; These fields hold the key or value associated with an entry Most of the time you won t need to use KeyValuePair<TK, TV> directly because SortedDictionary<TK, TV> allows you to work the keys and values individually However, when enumerating a SortedDictionary<TK, TV>, such as in a foreach loop, the objects being enumerated are KeyValuePairs In a SortedDictionary<TK, TV>, all keys must be unique, and a key must not change while it is in use as a key Values need not be unique Here is an example that demonstrates SortedDictionary<TK, TV> It reworks the Dictionary<TK, TV> example shown in the preceding section In this version, the database of employees and salaries is sorted based on name (which is the key)

// Demonstrate the generic SortedDictionary<TK, TV> class using System; using SystemCollectionsGeneric; class GenSortedDictionaryDemo { static void Main() { // Create a Dictionary that holds employee // names and their corresponding salary SortedDictionary<string, double> dict = new SortedDictionary<string, double>(); // Add elements to the collection dictAdd("Butler, John", 73000); dictAdd("Swartz, Sarah", 59000); dictAdd("Pyke, Thomas", 45000); dictAdd("Frank, Ed", 99000); // Get a collection of the keys (names) ICollection<string> c = dictKeys; // Use the keys to obtain the values (salaries) foreach(string str in c) ConsoleWriteLine("{0}, Salary: {1:C}", str, dict[str]); } }

(757a)

The output is shown here:

It turns out that, with a little notation, we can make this process both convenient and straightforward We now illustrate this new paradigm with some examples We begin with an inde nite integral

Butler, John, Salary: $73,00000 Frank, Ed, Salary: $99,00000 Pyke, Thomas, Salary: $45,00000 Swartz, Sarah, Salary: $59,00000

Ii(2 ) Ii(3) Ii(4 )

Part II:

.

(757b)

Math Note: Always be sure to check your work You can differentiate the answer in the last example to recover the integrand, con rming that the integration has been performed correctly EXAMPLE 79

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