Is either of the following equations correct? Give reasons for your answers.

a. (1/cosx) ∫ cos x dx = tan x + C
b. (1/cosx) ∫ cos x dx = tan x + C / cos x

검증된 단계별 안내

Start by evaluating the integral \( \int \cos x \, dx \). Recall that the integral of \( \cos x \) is \( \sin x + C \), where \( C \) is the constant of integration.

Rewrite the left side of both equations by substituting the integral: \( \frac{1}{\cos x} \int \cos x \, dx = \frac{1}{\cos x} (\sin x + C) \). This simplifies to \( \frac{\sin x}{\cos x} + \frac{C}{\cos x} \).

Recognize that \( \frac{\sin x}{\cos x} = \tan x \), so the expression becomes \( \tan x + \frac{C}{\cos x} \).

Compare this result with the right sides of the given equations: (a) \( \tan x + C \) and (b) \( \tan x + \frac{C}{\cos x} \). Notice that the constant term in (a) does not match the derived expression, while (b) matches exactly.

Conclude that equation (b) is correct because it properly accounts for the constant of integration divided by \( \cos x \), whereas equation (a) incorrectly treats the constant as a simple additive constant without division by \( \cos x \).

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Integration of Basic Trigonometric Functions

Understanding how to integrate basic trigonometric functions like cos x is essential. The integral of cos x with respect to x is sin x plus a constant of integration, C. This fundamental fact helps evaluate the given expressions correctly.

Algebraic Manipulation of Expressions

After integrating, it is important to correctly manipulate algebraic expressions, especially when dividing by functions like cos x. This includes distributing division over sums and understanding how constants of integration behave under division.

Properties of the Constant of Integration

The constant of integration, C, represents an arbitrary constant and must be handled carefully. When dividing an integral expression by a function, the constant remains arbitrary and should not be combined or altered incorrectly, as it represents an infinite family of antiderivatives.

Is either of the following equations correct? Give reasons for your answers.a. (1/cosx) ∫ cos x dx = tan x + Cb. (1/cosx) ∫ cos x dx = tan x + C / cos x