Chapter 5: Q 20. (page 451)

Consider the integral $鈭� s e c^{2} x \tan^{3} x d x$
(a) Solve this integral by using integration by substitution with u = tan x.
(b) Solve this integral another way, by using integration by substitution with u = sec x.

Short Answer

Expert verified

(a) $\begin{array}{rcl} \frac{\tan^{4} x}{4} + C \end{array}$

(b) $\begin{array}{rcl} \frac{\tan^{4} x}{4} + C \end{array}$

Step by step solution

Step 1. Given information

Integral is $鈭� s e c^{2} x \tan^{3} x d x$

Part (a) Step 1. Explanation

Let $u = \tan x$

$\begin{array}{rcl} 鈭� s e c^{2} x \tan^{3} x d x & = & 鈭� u^{3} d u \\ = & \frac{u^{4}}{4} + C \\ = & \frac{\tan^{4} x}{4} + C \end{array}$

Part (b) Step 1. Explanation

Let $u = \sec x$ .

$d u = s e c x \tan x d x$

$\begin{array}{rcl} 鈭� s e c^{2} x \tan^{3} x d x & = & 鈭� u (u^{2} - 1) d u \\ = & 鈭� u^{3} - u d u \\ = & \frac{u^{4}}{4} - \frac{u^{2}}{2} + C \\ = & \frac{s e c^{4} x}{4} - \frac{s e c^{2} x}{2} + C \\ = & \frac{\tan^{4} x}{4} + C \end{array}$

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91影视

Consider the integral $鈭� s e c^{2} x \tan^{3} x d x$
(a) Solve this integral by using integration by substitution with u = tan x.
(b) Solve this integral another way, by using integration by substitution with u = sec x.

Short Answer

Step by step solution

Step 1. Given information

Part (a) Step 1. Explanation

Part (b) Step 1. Explanation

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91影视

Consider the integral鈭�sec2xtan3xdx(a) Solve this integral by using integration by substitution with u = tan x.(b) Solve this integral another way, by using integration by substitution with u = sec x.

Short Answer

Step by step solution

Step 1. Given information

Part (a) Step 1. Explanation

Part (b) Step 1. Explanation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Applied Mathematics

Pure Maths

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Consider the integral $鈭� s e c^{2} x \tan^{3} x d x$
(a) Solve this integral by using integration by substitution with u = tan x.
(b) Solve this integral another way, by using integration by substitution with u = sec x.