What is the integral of tan(x) with respect to x? (2021)

What is the integral of tan(x) with respect to x? (2021)

Step 1: Recall the definition of the integral. We want to find the function whose derivative is tan(x).
Step 2: Remember that tan(x) can be rewritten as sin(x)/cos(x).
Step 3: Use the substitution method. Let u = cos(x). Then, the derivative of u is du = -sin(x) dx.
Step 4: Rewrite the integral in terms of u. The integral of tan(x) dx becomes -∫(1/u) du.
Step 5: The integral of 1/u is ln|u| + C. So, we have -ln|u| + C.
Step 6: Substitute back u = cos(x) into the equation. This gives us -ln|cos(x)| + C.
Step 7: Therefore, the integral of tan(x) with respect to x is -ln|cos(x)| + C.

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