Logarithm and trigonometry Is ln(sin x−cos x) equal to ln(cos x−sin x)?

Nigro6f

Nigro6f

Answered question

2022-10-15

Logarithm and trigonometry
Is ln ( sin x cos x ) equal to ln ( cos x sin x )?
So I did a integral problem but the answer is not same the answer given.
I'm given this question ( 2 1 tan x ) d x
So I got the answer which is x ln ( tan x 1 ) + 1 2 ln ( sec 2 x ) + c Then I simplify it into x ln ( sin x cos x ) + c
But the given answer is x ln ( cos x sin x ) + c

Answer & Explanation

fitte8b

fitte8b

Beginner2022-10-16Added 12 answers

No, but ln | sin x cos x | = ln | cos x sin x | . Both functions you're interested in have the same derivative, but are defined on different intervals. Using the absolute value encompasses both.
djo57bgfrqn

djo57bgfrqn

Beginner2022-10-17Added 4 answers

The antiderivative of 1 x is ln | x | , defined both for negative and positive values of the argument. So the sign is irrelevant.
As you can check,
x > 0 ( ln | x | ) = ( ln x ) = 1 x x < 0 ( ln | x | ) = ( ln ( x ) ) = 1 x = 1 x .

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