Exercise 8: We endeavor to solve the integral: dx 1+ sin x + cos x (a) Assume x E (-n, 7). Let u = (b) Using the expression for x in part (a), solve for dr in terms of u: it should be a rational function of tan(x/2). Solve for x. и. (c) Draw a triangle or use your favorite method to determine sin(x/2) and cos(x/2) in terms of u. cos² (x/2) – sin² (x/2) to solve for an expression for (d) Use sin(x) = 2 sin(x/2) cos(x/2) and cos(x) sin(x) and cos(x) in terms of u: they should, once again, be rational functions of u. (e) Perform these substitutions in the given integral: since sin(x), cos(x), and dx are all rational functions of u, the result should be a rational function. (f) Evaluate the resulting integral by partial fractions. (g) Back-substitute to get an expression for x.

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.