How do you Integrate tan (x)

Question – How do you integrate tan (x)?

Answer-

The result of integrating tan x is typically written as ln|sec x| + C. This is the standard result. It is possible to integrate the trigonometric function tan x, and the outcome of this integration is a formula that may be easily recalled. In the following paragraph, we will examine a solution to the integration problem involving tan x.

Tan X = Sin x/ Cos x

Integration of Tan X

In order to determine how to integrate tan x with respect to x, we first need to transform tan x into an integrable function by expressing it in terms of sine and cosine. According to the definition of tan x, we have tan x equal to sin x divided by cos x.

∫ tan x =∫ (sin x /cos x). dx

This can be rewritten as ∫1/cosx. sin x. dx

The substitution method of integration can be used to get the tan x indefinite integral.

∫ f(g(x)) g'(x) dx = ∫ f(u) du = F(u) + C

Let u = cos x. Then du = – sin x. dx

 = dx = – du/ sin x

∫ (sin x /cos x). dx = – ∫ du/u

Using the standard integration formula, we can determine that ∫ dx/x = ln x+ C

Thus ∫ (sin x /cos x). dx = – ∫ du/ u = – ln|u| + c

= -ln | (cos x) +C

= ln | (cos x)^-1 +C

= ln (sec x) + C

∫ (sin x /cos x). dx = ln (sec x) + C

∫ tan x = ln (sec x) + C

Hence the integration of tan x is ln|sec x| + C.

Also see: In Indian Rupees, 1 Trillion Is Equal To How Many Crores?