what is the integration of tan x

∫ tan x dx = −ln|cos x| + C = ln|sec x| + C.

Quick Scoop

Here’s the idea in simple steps:

1. Rewrite tan x.
   • Use the identity tan x = sin x / cos x.

2. Set up the integral.
   • ∫ tan x dx = ∫ (sin x / cos x) dx.

3. Use substitution.
   • Let u = cos x, then du = −sin x dx, so −du = sin x dx.

 * The integral becomes ∫ (sin x / cos x) dx = ∫ (1/u)(−du) = −∫ (1/u) du.

4. Integrate.
   • −∫ (1/u) du = −ln|u| + C.

 * Substitute back u = cos x → −ln|cos x| + C. 

5. Alternate form.
   • Because −ln|cos x| = ln|sec x|, you can also write the answer as:
     ∫ tan x dx = ln|sec x| + C.

So, the standard result used in calculus is:

• ∫ tan x dx = −ln|cos x| + C, or equivalently,
• ∫ tan x dx = ln|sec x| + C.

TL;DR: The integration of tan x is a logarithmic function: −ln|cos x| + C or ln|sec x| + C.

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