Chapter 5: Q. 4. (page 451)

Compile lists of
(a) the derivatives and (b) the integrals of the six basic trigonometric functions:
$\sin x, \cos x, \tan x, s e c x, c s c x, a n d c o t x .$

Short Answer

Expert verified

It is assumed that you are familiar with the following rules of differentiation.

$\begin{array}{rcl} D (\sin x) & = & \cos x \\ D (\cos x) & = & - \sin x \\ D (\tan x) & = & s e c^{2} x \\ D (s e c x) & = & s e c x \tan x \\ D (c s c x) & = & - c s c x c o t x \\ D (c o t x) n & = & - c s c^{2} x \end{array}$

These lead directly to the following indefinite integrals.

$\int \begin{array}{rcl} (\sin x) \end{array} d x \begin{array}{rcl} = \end{array} \begin{array}{rcl} - \end{array} \begin{array}{rcl} \cos \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} + \end{array} \begin{array}{rcl} c \end{array} \begin{array}{rcl} \int \end{array} \begin{array}{rcl} (\cos x) \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} \sin \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} + \end{array} \begin{array}{rcl} c \end{array} \begin{array}{rcl} \int \end{array} \begin{array}{rcl} (\tan x) \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} \log \end{array} \begin{array}{rcl} |(s e c x)| \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} + \end{array} \begin{array}{rcl} c \end{array} \begin{array}{rcl} \int \end{array} \begin{array}{rcl} (s e c x) \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} \log \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} |s e c x + \tan x| \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} + \end{array} \begin{array}{rcl} c \end{array} \begin{array}{rcl} \int \end{array} \begin{array}{rcl} (c s c x) \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} \log \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} |c s c x - c o t x| \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} + \end{array} \begin{array}{rcl} c \end{array} \begin{array}{rcl} \int \end{array} \begin{array}{rcl} (c o t x) \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} \log \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} |\sin x| \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} + \end{array} \begin{array}{rcl} c \end{array}$

Step by step solution

Step 1. Given Information.

Writte the derivatives and integrals of all trigonometric functions.

(a) Step 2. Derivatives of a trigonometric functions.

(b) Step 3. Integrals formulation.

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Compile lists of
(a) the derivatives and (b) the integrals of the six basic trigonometric functions:
$\sin x, \cos x, \tan x, s e c x, c s c x, a n d c o t x .$

Short Answer

Step by step solution

Step 1. Given Information.

(a) Step 2. Derivatives of a trigonometric functions.

(b) Step 3. Integrals formulation.

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Compile lists of(a) the derivatives and (b) the integrals of the six basic trigonometric functions:sinx,cosx,tanx,secx,cscx,andcotx.

Short Answer

Step by step solution

Step 1. Given Information.

(a) Step 2. Derivatives of a trigonometric functions.

(b) Step 3. Integrals formulation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Theoretical and Mathematical Physics

Applied Mathematics

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

Compile lists of
(a) the derivatives and (b) the integrals of the six basic trigonometric functions:
$\sin x, \cos x, \tan x, s e c x, c s c x, a n d c o t x .$