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Chapter 5: Q 29. (page 452)

Solve the following integral.
$\int c o t^{5} x d x$

Short Answer

Expert verified

Answer is $- \frac{c o t^{4} x}{4} + \frac{c o t^{2} x}{2} - \frac{\ln (c o t^{2} x + 1)}{2} + C$

Step by step solution

01

Step 1. Given information

An integral is $\int c o t^{5} x d x$

02

Step 2. Explanation

Let $c o t x = u$

\begin{array}{rcl} \int c o t^{5} x d x & = & \int \frac{- u^{5}}{u^{2} + 1} d u \\ = & - \int u^{3} - u + \frac{u}{u^{2} + 1} d u \\ = & - \frac{u^{4}}{4} + \frac{u^{2}}{2} - \frac{\ln (u^{2} + 1)}{2} + C \\ = & - \frac{c o t^{4} x}{4} + \frac{c o t^{2} x}{2} - \frac{\ln (c o t^{2} x + 1)}{2} + C \end{array}

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Most popular questions from this chapter

Consider the integral $\int \sin x \cos x d x$ .

(a) Solve this integral by using u-substitution with $u = \sin x$ and $d u = \cos x d x$ .

(b) Solve the integral another way, using u-substitution with $u = \cos x$ and $d u = - \sin x d x$ .

(c) How must your two answers be related? Use algebra to prove this relationship.

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

$u (x) = \sin x$

Solve given definite integral.

$\int_{1}^{2} \frac{3}{{(9 - 2 x^{2})}^{3 / 2}} d x$

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.

$\int \frac{x^{3} - 3 x^{2} + 2 x - 3}{x^{2} + 1}$ dx

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 $\int \frac{1}{\sqrt{x^{2} - 4}} d x$ .

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

(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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