Integral Of Tan X

Introduction about integral:

Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. The term integral may also refer to the notion of anti derivative, a function F whose derivative is the given function .

[Source: Wikipedia]

Example 1:

Integral of tanx:

Solution:

We know that in trigonometry tan x = sin x/cos x

'int tan x dx' = 'int sinx / cos x dx'

Let us take,
u = cos x.
then differentiate,

du = - sin x dx

Plug du=-sin x, u=cos x in integral

'int sinx / cos x dx' = (-1)'int (-1)sinx / cos x dx'

'int - (du)/u'


Solve the integral

= - ln |u| + C

Plug back u=cos x

= - ln |cos x| + C

Another result for Integral of tan x:

tan x dx = - ln |cos x| + C
= ln | (cos x)-1 | + C
= ln |sec x| + C

Therefore:

tan x dx = - ln |cos x| + C = ln |sec x| + C

Example 2:

Integral of (tan (2x)) 2

Solution:

'int' (tan 2x)2 dx

We know that 1+ tan2x = sec2x

Use this trigonometry formula we can write as,

(tan 2x)2 = (sec 2x)2-1

Plug in the integration

'int' [(sec 2x)2 1] dx

From the integration formula 'int'sec2 x = tan x

'int' [(sec 2x)2 1] dx = [tan 2x] x + C

The answer is = [tan 2x] x +C

Example 3:

Integral of (tan 2 x)

Solution:

'int' (tan2 x) dx

We know that 1+ tan2x = sec2x

Use this trigonometry formula we can write as,

(tan2 x) = (sec2 x)-1

Now substitute this value in integration

'int' (tan2 x) dx = 'int'(sec2 x - 1) dx

From the integration formula 'int'sec2 x = tan x

'int' (sec2 x - 1) dx = tan x x + C

The answer is tan x x+ C