Solving the Integral of Cosine Function or Finding the Floor Value of Cos(x): Explained and Clarified

int cosx

The expression “int cosx” is not clear and could have two different interpretations, depending on the intended meaning

The expression “int cosx” is not clear and could have two different interpretations, depending on the intended meaning.

1. If “int” is a shorthand for the integration symbol (integral), and “cosx” represents the cosine function, then:

∫cos(x) dx represents the integral of the cosine function with respect to x. To solve this integral, we can use the trigonometric identity cos^2(x) + sin^2(x) = 1, which can be rewritten as cos(x) = √(1 – sin^2(x)). By substituting this into the integral, we get:

∫√(1 – sin^2(x)) dx

This integral can be solved using a trigonometric substitution, by letting x = sin(θ), dx = cos(θ) dθ:

∫√(1 – sin^2(x)) dx = ∫√(1 – sin^2(θ)) cos(θ) dθ

Using the Pythagorean identity cos^2(θ) + sin^2(θ) = 1, we have cos(θ) = √(1 – sin^2(θ)):

∫cos(θ) dθ = ∫cos(θ) dθ

The integral on the right side is straightforward to solve, and the result is simply:

∫cos(θ) dθ = sin(θ) + C

Substituting back x = sin(θ), we obtain the final result:

∫cos(x) dx = sin(x) + C

2. If “int” is meant to represent the integer part or floor function, then “int cosx” could mean the largest integer less than or equal to the value of cos(x). For example, if x = π/4, then cos(x) = √2/2, and the largest integer less than or equal to √2/2 is 0. Therefore, “int cosx” in this context would be 0.

However, without further context or clarification, it is difficult to definitively determine the intended meaning of “int cosx.”

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