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Chapter 5: Q 28. (page 452)

Solve the following integral.
$\int \tan^{6} x d x$

Short Answer

Expert verified

Answer is $\begin{array}{rcl} \frac{\tan^{5} x}{5} - \frac{\tan^{3} x}{3} + \tan x - x + C \end{array}$

Step by step solution

01

Step 1. Given information

An integral is $\int \tan^{6} x d x$

02

Step 2. Explanation

Let $\tan x = u$

$\begin{array}{rcl} \int \tan^{6} x d x & = & \int \frac{u^{6}}{u^{2} + 1} d u \\ = & \int u^{4} - u^{2} + 1 - \frac{1}{u^{2} + 1} d u \\ = & \frac{u^{5}}{5} - \frac{u^{3}}{3} + u - \tan^{- 1} u + C \\ = & \frac{\tan^{5} x}{5} - \frac{\tan^{3} x}{3} + \tan x - x + C \end{array}$

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Most popular questions from this chapter

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 $u$ .

Describe two ways in which the long-division algorithm for polynomials is similar to the long-division algorithm for integers and then two ways in which the two algorithms are different.

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.

Solve the integral: $\int \ln \sqrt{x} d x$

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.

$\int x \sqrt{x^{2} + 1} d x$

See all solutions

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