Integral tan 2x

In mathematics, Integration is one of the concepts in calculus. Integration is a limiting process of finding integral of a function and also used to find the area of a region under a curve. We can also say that integral is an anti derivative of differentiation. Integration of a function is shown as,

f(x) dx = F(x) + C

`int` = Sign of integration

The variable x in dx is called integrator.

C= constant.

Integral tan 2x:

The standard formula to find the integral of tanx is ,

tan x dx = log

sec x

+C

Using this formula we will solve the problems related to Integral tan 2x in the following section.

Example problems on Integral tan 2x:

Integral tan 2x Example 1:

Find the integral of tan2x

Solution:

`int` tan (2x) dx

Let t = 2x dt = 2 dx

then ,

`1/2` dt = dx

Now find the integral tan2x with the replacement of t ,

1/2 ? tan(t) du

Integrate:

`1/2 ` ? tan (t) du = `1/2` [log

sec t

+C { By the formula }

Now substitute t= 2x

? tan (2x) dx = `1/2` [log

sec 2x

+C]

Integral tan 2x Example 2:

Find the integral of tan2 2x

Solution:

? tan2 2x dx

By formula,

Sec2x= 1+tan2x

Hence tan2x = Sec2x -1

Let t= 2x

dt= 2 dx

½ dt = dx

? tan2 2x dx = `1/2` ? tan2 t dt

= `1/2` ? (Sec2t -1) dt

= `1/2` { ? Sec2t dt - ? 1 dt }

= `1/2` `( tan t)/2 ` – ( ` t /2` )

Now replace t as 2x ,

? tan2 2x dx =`1/2` [ tan(2x) –2x ]

Hence the answer is `(tan2x-2x)/2`

Integral tan 2x Example 3:

Find the integral of tan2x+ x

Solution:

? tan2x+ x dx this can be written as,

? tan2x+x = ?tan2x dx +? x dx

Step 1: Find integral tan2x

?tan2x dx

Let t = 2x dt = 2 dx

then, `1/2` dt = dx

Now find the integral tan2x with the replacement of t ,

`1/2` ? tan(t) du

Integrate:

`1/2 ` ? tan (t) du = `1/2` [log

sec t

+C { By the formula }

Now substitute t= 2x

? tan (2x) dx = `1/2` [log

sec 2x

+C]

Step 2: Find ? x dx

? x dx = x2/2

The final solution is ,

? tan2x+ x dx =`1/2` [log

sec 2x

] + `(x^2)/2`

Now take `1/2` out as a common term,

Hence the answer is `(log(sec 2x)+(x^2))/2`

Integration, a field of mathematics, which deals with the limits of certain functions from where to where it exists. Integration is the field of math used to solve the functions with respect to their limits. Each function consist of two limits, they represent the starting and ending points of the curve. The integral solution justifies the existence of a curve. In integration animation article, you can learn about the integral solving mechanism.

` F(y) = ``int_a^bf(y)dy`

Where, a = lower limit,

b = upper limit. Solved Examples - integration animation:

The below are the solved examples of integration animation article.

Example 1:

Find the value of the integration

`int_0^4(3y)dy`

Solution:

Integrate with respect to y, we get

`int_0^4(3y)dy` = 3 (y2 / 2)40

Substitute the limits, we get

= 3 (42 / 2) - 3 (02 / 2)

= 3 (8) - 3 (0)

= 24.

Solution:

answer: 24

Example 2:

Find the value of the integration

`int_1^4(y^3)dy`

Solution:

Integrate with respect to y, we get

`int_1^4(y^3)dy` = (y4 / 4)41

Substitute the limits, we get

= (44 / 4) - (14 / 4)

= (256 / 4) - (1 / 4)

= (255 / 4)

= 63.75

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