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Definite integral of absolute value function | AP Calculus AB | Khan Academy
Definite integral of absolute value function | AP Calculus AB | Khan Academy

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Integration of Tan X

The standard result of the integration of tan x is ln|sec x| + C. The trigonometric function tan x is integrable and this standard result of the integration of tan x is remembered as a formula. Let us learn how to solve the integration of tan x in the upcoming section.

1. What is Integration of Tan X?
2. How to Solve Integration of Tan X?
3. Definite Integration of Tan X
4. Graph of Integration of Tan X
5. FAQs on Integration of Tan X

What Is Integration of Tan X?

The integration of tan x is -ln|cos x| + C (or) ln|sec x| + C. The function f(x) = tan x is continuous at all real numbers, except x = (2n+ 1)π/2, The domain of the function = range of the function tan(x), except for the odd multiples of π/2. Hence tan x is integrable except for that interval with respect to x. We do the integration of tan x by the integration by substitution.

How to Solve Integration of Tan X?

To find the integration of tan x, with respect to x, we express tan x in terms of sine and cosine so that it becomes an integrable function. As per the definition of tan x, we have tan x = sin x / cos x

∫ tan x =∫ (sin x /cos x) .dx

This can be rewritten as \(\int \dfrac{1}{\cos x}\). sin x. dx

Let us find the indefinite integral of tan x using the substitution method of integration.

∫ f(g(x)) g'(x) dx = ∫ f(u) du = F(u) + C

Let u = cos x. Then du = – sin x . dx

⇒ dx = – du/ sin x

∫(sin x /cos x). dx = – ∫ du/ u

By the standard integration formula, we know that ∫ dx/x = ln x+ C

Thus ∫ (sin x /cos x) .dx = – ∫ du/ u = – ln|u| + c

= -ln |(cos x)+C

= ln |(cos x) -1+C

= ln (sec x) + C

∫ (sin x /cos x) .dx = ln (sec x) + C

∫ tan x = ln (sec x) + C

Thus the integration of tan x is ln|sec x| + C.

Definite Integration of Tan x

By the definition of the fundamental theorems of definite integrals, we can compute the definite integration of tan x between any two intervals. Let us compute the integration of tan x between π/6 and π/3.

We apply the formula of definite integrals \(\int\limits_a^b f(x) dx\) = f(b) – f(a).

We know by the indefinite integration of tan x = -ln|cos x| + C. Here we take the absolute value only by computing the definite integrals.

Thus \(\int\limits_\dfrac{\pi }{3}^\dfrac{\pi }{6} tan(x) dx\) =

=-ln|cos x|\(^{\pi/2}_0\)

ln (cos \(\dfrac{\pi }{3}\)) – ln (cos \(\dfrac{\pi }{6}\))

= ln ½ – ln √3/2

Evaluating this further, we get lg √3 = ½ ln 3

Graph of Integration of Tan X

Let us evaluate the area under the graph tan x between 0 and π/2.

To find the \(\int\limits_0^\dfrac{\pi }{2}\)tan x dx, we apply the formula of definite integrals \(\int\limits_a^b f(x) dx\) = f(b) – f(a).

\(\int\limits_0^\dfrac{\pi}{2}\)tan x dx

= ln|sec x|\(^{\pi/2}_0\)

= ln|sec π/2| – ln|sec 0|

=ln(∞)- ln(1)

= ∞

Thus the graph of the integral of tan x diverges to infinity in the interval[0,π/2].

☛ Also Check

Examples of Integration of Tan X

  1. Example 1. Solve the integration of (tan x)2

    Solution:

    To find the integral of (tan x)2

    (tan x)2 = tan2 x

    = sec2 x – 1 (by the known trigonometric identity)

    ∫ tan2 x. dx= ∫ [sec2 x – 1] dx

    = ∫ (sec2 x). dx -\(\int\) dx

    Using the standard integration formula, we get

    = tan x -x + c

    Answer: The integration of (tan x)2 = tan x -x + c

  2. Example 2. What is the integration of tan (x/2) with respect to x?

    Solution:

    To find ∫ tan (x/2)

    We know that integration of tan x = -ln|cos x|+ C

    Applying, this we get ∫ tan (x/2) = -2 ln|cos x/2| + C

    -2 ln|cos x/2| = -ln |cos2 x/2| + C

    = ln|sec2 x/2| + C

    (or) = ln|1 + tan2 x/2| + C

    Answer: The integration of tan (x/2) = ln|sec2 x/2| + C (or) ln|1 + tan2 x/2| + C

  3. Example 3. Evaluate the integration of tan x in the interval 0 to π/4.

    Solution:

    Given f(x) = tan x

    The integration of tan x = log |sec x|

    To find the \(\int\limits_0^\dfrac{\pi }{4}\)tan x dx, we apply the formula of definite integrals \(\int\limits_a^b f(x) dx\) = f(b) – f(a).

    \(\int\limits_0^\dfrac{\pi}{4}\)tan x dx

    = ln|sec π/4| – ln|sec 0|

    =ln|√2|- ln|1|

    = ln√2- 0

    = ln√2

    = ln 2 1/2 = ½ ln 2

    Answer: The integration of tan x in the interval 0 to π/4 = ½ ln 2

Practice Questions on Integration of Tan X

FAQs on Integration of Tan X

What is Integration of Tan X?

The integration of tan x is ln|sec x| + C (or) -ln|cos x| + C.

Is Tan x Integrable?

Yes, Tan x is integrable. Tan x is a continuous function on its domain. The integration of tan x is -ln|cos x| + C.

How to do Integration of Tan X?

The integration of tan x is done by the method of integration by substitution. Tan x = sin x / cos x. Taking cos x as u, we get du = -sin x dx. ∫ tan x = ∫ (sin x /cos x) .dx

=-∫ du/ u = -ln u + C

= -ln|cos x| + C.

Thus ∫ tan x = = -ln|cos x| + C.

What is Integration of 2Tan X?

The integration of tan x is -ln|cos x| + C. Thus \(\int\) 2 tan x = 2 \(\int\) tan x

∫ 2 tan x = -2 ln|cos x| + C.

= – ln|cos2 x| + C.

Is the Differentiation and Integration of Tan x the Same?

No. the differentiation and integration of tan x are not the same. The differentiation of tan x is sec2 x and the integration of tan x is ln|sec x| + C.

What is The Technique We Use To Find The Integration of Tan X?

the integration of tan x is done by the method of integration by u-substitution. We write tan x in the integrable form sin x / cos x and then take u(x) is cos(x).

By the method of substitution, we know that ∫ f(g(x)) g'(x) dx =∫ f(u) du = F(u) + C, where g(x) = f(u). We apply this u-substitution technique for the integration of tan x and arrive at the standard result as ∫ tan x = log |sec x|

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