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then we would see that we are integrating an expression of the form ( x) ( x) (which we in fact encountered among our differentiation rules in Section 25) As we know, expressions like this arise from differentiating log ( x) Returning to the original problem, we pose our initial guess as log[x 2 + 3] Differentiation of this expression gives the answer 2x/[x 2 + 3] This is close to what we want, but we must adjust by a factor of 1/2 We write our final answer as 1 x dx = log[x 2 + 3] + C 2 +3

Part I:

Although it is not wrong to use two where clauses as just shown, the same effect can be achieved in a more compact manner by using a single where in which both tests are combined into a single expression Here is the query rewritten to use this approach:

var posNums = from n in nums where n > 0 && n < 10 select n;

.

In general, a where condition can use any valid C# expression that evaluates to a Boolean result For example, the following program defines an array of strings Several of the strings define Internet addresses The query netAddrs retrieves only those strings that have more than four characters and that end with net Thus, it finds those strings that contain Internet addresses that use the net top-level domain name

We observe that the expression 6x 2 + 4x is nearly the derivative of x 3 + x 2 + 1 In fact if we set ( x) = x 3 + x 2 + 1 then the integrand (the quantity that we are asked to integrate) is [ ( x) ]50 2 ( x) It is natural to guess as our antiderivative [ ( x) ]51 Checking our work, we find that [ ( x) ]51 = 51[ ( x) ]50 ( x)

// Demonstrate another where clause using System; using SystemLinq; class WhereDemo2 { static void Main() { string[] strs = { "com", "net", "hsNameAcom", "hsNameBnet", "test", "network", "hsNameCnet", "hsNameDcom" }; // Create a query that obtains Internet addresses that // end with net var netAddrs = from addr in strs where addrLength > 4 && addrEndsWith("net", StringComparisonOrdinal) select addr; // Execute the query and display the results foreach(var str in netAddrs) ConsoleWriteLine(str); } }

The output is shown here:

Notice that the program makes use of one of string s methods called EndsWith( ) within the where clause It returns true if the invoking string ends with the character sequence specified as an argument

We see that the answer obtained is quite close to the answer we seek; it is off by a numerical factor of 2/51 With this knowledge, we write our final answer as ( x 3 + x 2 + 1) 50 ( 6x 2 + 4x) dx = 2 [x 3 + x 2 + 1]51 + C 51

Often you will want the results of a query to be sorted For example, you might want to obtain a list of past-due accounts, in order of the remaining balance, from greatest to least Or, you might want to obtain a customer list, alphabetized by name Whatever the purpose, LINQ gives you an easy way to produce sorted results: the orderby clause

19:

You can use orderby to sort on one or more criteria We will begin with the simplest case: sorting on a single item The general form of orderby that sorts based on a single criterion is shown here: orderby sort-on how The item on which to sort is specified by sort-on This can be as inclusive as the entire element stored in the data source or as restricted as a portion of a single field within the element The value of how determines if the sort is ascending or descending, and it must be either ascending or descending The default direction is ascending, so you won t normally specify ascending Here is an example that uses orderby to retrieve the values in an int array in ascending order:

// Demonstrate orderby using System; using SystemLinq; class OrderbyDemo { static void Main() { int[] nums = { 10, -19, 4, 7, 2, -5, 0 };