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Chapter 5: Q 11. (page 451)

Describe strategy for solving the type of integral given.
$\int c o t^{k} x d x, k$ even.

Short Answer

Expert verified

Write $c o t x = u$

Step by step solution

Step 1. Given information

Integral is $\int c o t^{k} x d x$

Step 2. Explanation

Consider the integral $\int c o t^{k} x d x, k$ even

Let $c o t x = u$ and use the formula $(c o t x)' = - c s c^{2} x$

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Most popular questions from this chapter

Solve given definite integral.

$\int_{4}^{5} \frac{1}{x \sqrt{x^{2} + 9}} d x$

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$\int \frac{x}{\sqrt{x^{2} - 1}} d x$

Why is it okay to use a triangle without thinking about the unit circle when simplifying expressions that result from a trigonometric substitution with $x = a \sin u$ or $x = a \tan u$ ? Why do we need to think about the unit circle after trigonometric substitution with $x = a s e c u$ ?

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 $u$ .

Solve $\int \frac{x}{4 + x^{2}} d x$ the following two ways:

(a) with the substitution $u = 4 + x^{2};$

(b) with the trigonometric substitution x = 2 tan u.

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Describe strategy for solving the type of integral given.∫cotkxdx,keven.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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Describe strategy for solving the type of integral given.
$\int c o t^{k} x d x, k$ even.