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Integral of tan^2x
Integral of tan^2x

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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.

1. What is Integration of Tan Square x?
2. Integration of Tan Square x Formula
3. Integration of Tan Square x Proof
4. Integration of Tan Square x From 0 to Pi by 4
5. FAQs on Integration of Tan Square x

What is Integration of Tan Square x?

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.

Integration of Tan Square x Formula

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:

Integration of Tan Square x Proof

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.

Integration of Tan Square x From 0 to Pi by 4

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:

Integration of Tan Square x Examples

  1. Example 1: Find the integration of tan square x sec square x.

    Solution: We will find the integration of tan square x sec square x using the substitution method.

    Assume u = tan x ⇒ du = sec2x dx

    ∫tan2x sec2x dx = ∫u2 du

    = u3/3 + C

    = (1/3) tan3x + C

    Answer: ∫tan2x sec2x dx = (1/3) tan3x + C

  2. Example 2: Evaluate the integration of tan square x minus cot square x.

    Solution: To find the integration of tan square x minus cot square x, we will use trigonometric identities

    • 1 + tan2x = sec2x
    • 1 + cot2x = cosec2x

    We have,

    ∫(tan2x – cot2x) dx = ∫[(sec2x – 1) – (cosec2x – 1)] dx

    = ∫(sec2x – 1 – cosec2x + 1) dx

    = ∫(sec2x – cosec2x) dx

    = ∫sec2x dx + ∫(- cosec2x) dx

    = tan x + cot x + C [Because d(tan x)/dx = sec2x and d(cot x)/dx = – cosec2x]

    Answer: Hence, ∫(tan2x – cot2x) dx = tan x + cot x + C

Integration of Tan Square x Questions

FAQs on Integration of Tan Square 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.

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