What is the integral of tan(x) dx? (2023)

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Question: What is the integral of tan(x) dx? (2023)

Options:

Correct Answer: -ln

Solution:

The integral of tan(x) is -ln|cos(x)| + C.

What is the integral of tan(x) dx? (2023)

What is the integral of tan(x) dx? (2023)

Step 1: Recall the definition of the tangent function: tan(x) = sin(x) / cos(x).
Step 2: Rewrite the integral: ∫tan(x) dx = ∫(sin(x) / cos(x)) dx.
Step 3: Use a substitution method. Let u = cos(x). Then, the derivative du = -sin(x) dx.
Step 4: Rearrange the substitution: dx = -du / sin(x).
Step 5: Substitute u and dx into the integral: ∫(sin(x) / u)(-du / sin(x)) = -∫(1/u) du.
Step 6: The integral of 1/u is ln|u|, so we have -ln|u| + C.
Step 7: Substitute back u = cos(x): -ln|cos(x)| + C.

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