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Chapter 5: Q 30. (page 452)

Solve the following integral.
$\int c s c^{2} x d x$

Short Answer

Expert verified

Answer is $- c o t x + C$

Step by step solution

Step 1. Given information

An integral is $\int c s c^{2} x d x$

Step 2. Explanation

$\int c s c^{2} x d x = - c o t x + C$

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Most popular questions from this chapter

For each function u(x) in Exercises 9–12, write the differential du in terms of the differential dx.

$u (x) = \sin x$

Suppose $u (x) = x^{2}$ . Calculate and compare the values of the following definite integrals:

role="math" localid="1648786835678" $\int_{- 1}^{5} u^{2} d u, \int_{x = - 1}^{x = 5} u^{2} d u and \int_{u (- 1)}^{u (5)} u^{2} d u$

Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.

$\int \frac{x^{4} - 3}{2 + 3 x^{2}} d x$

Show that if $x = \tan u$ , then $d x = \sec^{2} u d u$ , in the following two ways: (a) by using implicit differentiation, thinking of $u$ as a function of $x$ , and (b) by thinking of $x$ as a function of $u$ .

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{x^{2} - 4} d x$ .

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{\sqrt{x^{2} - 4}} d x$ .

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving $\int {(x^{2} + 4)}^{- 5 / 2} d x$

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form $\sqrt{x^{2} - a^{2}}$ .

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with $x = a \sin u$ , we must consider the cases $x > a$ and $x < - a$ separately.

(h) True or False: When using trigonometric substitution with $x = a s e c u$ , we must consider the cases $x > a$ and $x < - a$ separately.

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Solve the following integral.∫csc2xdx

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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Solve the following integral.
$\int c s c^{2} x d x$