This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Let's solve this. To find the integral of tan x, we can rewrite tan x as (sin x)/(cos x). Step 1: Set up the integral. ∫ tan x dx = ∫ (sin x)/(cos x) dx Step 2: Use a substitution. Let u = cos x. Then, du = -sin x dx, which means sin x dx = -du. Step 3: Substitute u and du into the integral. ∫ (sin x)/(cos x) dx = ∫ (1)/(u) (-du) = -∫ (1)/(u) du Step 4: Integrate with respect to u. -∫ (1)/(u) du = -ln|u| + C Step 5: Substitute back cos x for u. -ln|u| + C = -ln|cos x| + C Step 6: Use logarithm properties to simplify. Since -ln|cos x| = ln|(cos x)⁻¹| = ln|(1)/(cos x)| = ln|sec x|. So, the integral is ln|sec x| + C. The integral of tan x is ln|sec x| + C. That's 2 down. 3 left today — send the next one.