Integration of Tan Square x

The formula for the integration of tan square x is tan x - x + C, with C as the integration constant. As we know that integration is nothing but the reverse process of differentiation, we can say that integration of tan square x is the same as the antiderivative of tan square x. We can determine the integral of tan²x using the trigonometric identity or by writing tan x in terms of sin x and cos x. Mathematically, we write the integration of tan square x as ∫ tan²x dx = tan x - x + C.

Let us calculate the integration of tan square x, determine its formula and the definite integral of tan square x with different limits. We will also solve some examples determining the integration of tan square x combined with some other trigonometric functions for a better understanding of the concept.

Integration of tan square x is the process of reverse differentiation of tan square x and finding its antiderivative. To find the integral of tan square x, we can use the trigonometric identities such as tan x = sin x/cos x and 1 + tan²x = sec²x. Mathematically, we can write the integration of tan square x as ∫ tan²x dx = tan x - x + C, where ∫ is the symbol of integration and C is the integration constant. Let us now go through the formula of the integration of tan square x.

Now, let us understand the formula of integral of tan²x. We can write the integration of tan square x mathematically as ∫ tan²x dx = tan x - x + C, where C is the integration constant. We will understand how to get this formula further in this article by proving the integral of tan square x. The integration of tan square x is the tangent of angle x minus x plus the constant of integration which is given symbolically in the image below:

Now we know that the integration of tan square x is tan x - x + C which we will prove now using trigonometric identity sin²x + cos²x = 1. We will write tan x as sin x/ cos x. Therefore, we have

∫ tan²x dx = ∫ (sin x/ cos x)² dx

= ∫ (sin²x / cos²x) dx

= ∫ (1 - cos²x)/cos²x dx [Because sin²x + cos²x = 1 ⇒ sin²x = 1 - cos²x]

= ∫ (1/cos²x) dx - ∫ 1 dx

= ∫ sec²x dx - ∫ dx

= tan x - x + C [Because the derivative of tan x is sec²x ⇒ integral of sec²x is tan x + K]

We can also prove the integration of tan square x using the trigonometric identity 1 + tan²x = sec²x. To evaluate, we have

∫ tan²x dx = ∫ (sec²x - 1) dx

= ∫ sec²x dx - ∫1dx

= tan x - x + C

Hence, we have proved that the integration of tan square x is tan x - x + C.

We know that the formula for the integration of tan square x is tan x - x + C. Next, we will find its definite integral with limits from 0 to π/4 using this formula. For this, we have

\(\begin{align}\int_{0}^{\frac{\pi}{4}} \tan^2x \ dx&=\left [ \tan x - x + C \right ]_0^\frac{\pi}{4}\\&=\tan \frac{\pi}{4} - \frac{\pi}{4}+C - \tan 0+0-C\\&=1-\frac{\pi}{4}\end{align}\)

Hence the integration of tan square x from 0 to pi by 4 is equal to 1 - π/4.

Important Notes on Integration of Tan Square

• The formula for the integration of tan square x is tan x - x + C.
• We can determine the integral of tan²x using the trigonometric identities such as sin²x + cos²x = 1 and 1 + tan²x = sec²x.
• The integration of tan square x from 0 to pi by 4 is equal to 1 - π/4.

☛ Related Topics:

• Integral of sec x tan x
• Integral of sin inverse x
• Integral of arctan
• Integration of sin x cos x

What is the Integration of Tan Square x in Calculus?

Integration of tan square x is the process of reverse differentiation of tan square x and finding its antiderivative. Mathematically, we can write the integration of tan square x as ∫ tan²x dx = tan x - x + C.

What is the Formula for Integration of Tan Square x?

The formula for the integration of tan square x is tan x - x + C, with C as the integration constant.

How to Find Integration of Tan Square x?

We can determine the integral of tan²x using the trigonometric identities such as sin²x + cos²x = 1 and 1 + tan²x = sec²x.

What is Integration of Tan Square x Sec Square x?

The integration of Tan Square x Sec Square x is given by, ∫tan²x sec²x dx = (1/3) tan³x + C.

How to Find the Integration of Tan Square x Minus Cot Square x?

The Integration of Tan Square x Minus Cot Square x is written as ∫(tan²x - cot²x) dx = tan x + cot x + C.

What is the Integration of Tan Square x Minus Sin Square x?

The Integration of Tan Square x Minus Sin Square x is given by, ∫(tan²x - sin²x) dx = tan x - (3/2)x + (1/4) sin 2x + C.