Integral of Tan 2x

Integral of tan 2x can be calculated using the substitution method and by expressing tan 2x in terms of sin and cos. Let us recall the concept of integral and trigonometric function tan 2x before getting to the integral of tan 2x. The integral of a function is the reverse process of differentiation and tan 2x is an important trigonometric formula. The integral of tan 2x is given by (-1/2)ln |cos 2x| + C.

In this article, we will derive the integral of tan 2x by the substitution method, determine the definite integral of tan 2x from 0 to π/4 along with solved examples for a better understanding.

The integral of tan 2x is (-1/2)ln |cos 2x| + C which can be calculated using the substitution method followed by integral of tan x formula or by expressing tan 2x in terms of sin and cos. Mathematically, the integral of tan 2x is written as ∫tan 2x dx = (-1/2)ln |cos 2x| + C, where C is the constant of integration, dx shows the integral of tan 2x is with respect to x and ∫ is the sign of integration.

The formula for integral of tan 2x is given by, ∫tan 2x dx = (-1/2)ln |cos 2x| + C, where C is the constant of integration.

Now, that we know the integral of tan 2x, let us derive the formula for the integral of tan 2x using the substitution method. We will use the following trigonometric and integration formulas to prove the integral of tan 2x:

• tan x = sin x/cos x
• ∫(1/x) dx = ln |x| + C

∫tan 2x dx = ∫sin 2x/cos 2x dx --- (1)

Assume cos 2x = u

Differentiating both sides of cos 2x = u, we get

-2 sin 2x dx = du ⇒ sin 2x dx = -du/2

Substitute sin 2x = -du/2 in (1),

∫tan 2x dx = ∫sin 2x/cos 2x dx

= ∫(-1/2u) du

= (-1/2) ∫(1/u) du

= (-1/2) ln |u| + C

= (-1/2) ln |cos 2x| + C

Hence, we have derived the integral of tan 2x formula using the substitution method. Next, we will derive the formula for the integral of tan 2x using the formula for the integral of tan x.

Now, we will prove the integral of tan 2x formula using the formula for integral of tan x. Assume 2x = u, differentiation 2x = u, we have 2dx = du which implies dx = du/2. Therefore, we have

∫ tan 2x dx = (1/2) ∫tan u du

= (1/2) [-ln |cos u| + C] {Using ∫tan x dx = -ln |cos x| + C}

= (1/2) [-ln |cos 2x| + C]

= (-1/2) ln |cos 2x| + K, where K = (1/2)C

Hence, we have derived the integral of tan 2x formula.

Next, we will evaluate the definite integral of tan 2x with limits from 0 to π. We know that the integral of tan 2x is given by, ∫tan 2x dx = (-1/2) ln |cos 2x| + C, where C is the constant of integration.

\(\begin{align}\int_{0}^{\pi} \tan 2x dx&=\left [ \dfrac{-1}{2}\ln|\cos 2x| + C \right ]_0^{\pi}\\&=(\dfrac{-1}{2}\ln|\cos2\pi| + C)-(\dfrac{-1}{2}\ln|\cos 0| + C)\\&=(\dfrac{-1}{2}\ln1 + C)-(\dfrac{-1}{2}\ln1 + C)\\&=0\end{align}\)

Hence, the integral of tan 2x from 0 to π is equal to 0.

Important Notes on Integral of Tan 2x

• The integral of tan 2x can also be written as ∫ tan 2x dx = (1/2)ln |sec x| + C = (-1/2) ln |cos 2x| + C
• The integral of tan 2x can be calculated by expressing tan 2x in terms of sin and cos

Topics Related to Integral of Tan 2x

• Integral Calculus
• Integral of Sin x

What is Integral of Tan 2x in Calculus?

The integral of tan 2x is (-1/2)ln |cos 2x| + C which can be calculated using the substitution method followed by integral of tan x formula or by expressing tan 2x in terms of sin and cos.

How to Find the Integral of Tan 2x?

The integral of Tan 2x can be calculated using the substitution method.

What is the Integral of Tan^2x?

The integral of tan^2x, that is, ∫tan²x dx is tan x - x + C.

What is the Integral of Tan 2x Formula?

The formula for the integral of tan 2x is (-1/2)ln |cos 2x| + C

Is the Integral of Tan 2x Equal to the Anti-derivative of Tan 2x?

Yes, the integral of tan 2x is equal to the anti-derivative of tan 2x as the integral is nothing but the anti-derivative of the function.

What is the Integral of Tan 2x from 0 to π?

The integral of tan 2x from 0 to π is equal to 0.

What is the Integral of Tan 2x in Terms of Sec?

The integral of tan 2x can also be written as ∫ tan 2x dx = (1/2)ln |sec x| + C