Solved Examples on Integral of Tan x

Example 1: Calculate the integral of cot x with respect to x.

Solution:

∫ cot x dx = ∫ cos x / sin x dx

Substitute sin x = t.

Differentiating with respect to x.

Then cos x dx = dt.

Then the above integral becomes

= ∫ (1/t) dt

= ln |t| + C (Because ∫ 1/x dx = ln|x| + C)

Substitute back t = sin x back here,

= ln |sin x| + C

Thus, ∫ cot x dx = ln |sin x| + C.

Example 2: Calculate the integral of sec x tan x with respect to x.

Solution:

We can write sec x = 1/cos x and tan x = sin x/cos x.

We have, ∫sec x tan x dx = ∫(1/cos x)(sin x/cos x) dx

= ∫(sin x/cos²x) dx

Now, assume cos x = t.

Differentiating both sides

-sin x dx = dt

⇒ sin x dx = dt

Substituting back these values

We have, ∫sec x tan x dx = ∫(-1/t²) dt

= (1/t) + C

Substitute back t = sin x back here,

= 1/cos x + C

= sec x + C

Thus, ∫ sec x tan x dx = sec x + C

Example 3: Calculate the integral of tan²x with respect to x.

Solution:

Let us find the integral of (tan² x) with respect to x.

= ∫ tan² x dx

Using the identity sec² X – tan² X = 1,

∫ tan² x dx = ∫ (sec² x – 1) dx

= ∫ sec² x dx – ∫ 1 dx

Using the standard integration formulas, ∫sec² x dx = tan x + C and ∫ 1 dx = x + C,

we get, tan x – x + C

Hence, ∫ tan² x dx = tan x – x + C.

Example 4: Calculate the integral of tan^-1 x with respect to x.

Solution:

We know that,

Formula for integration by parts is ∫f(x)g(x)dx = f(x) ∫g(x)dx – ∫[d(f(x))/dx × ∫g(x) dx] dx.

∫tan^-1x dx = ∫tan^-1x.1 dx

= tan^-1x ∫1dx – ∫[d(tan^-1x)/dx × ∫1 dx] dx

= x tan^-1x – ∫[1/(1 + x²) × x] dx

= x tan^-1x – ∫x/(1 + x²) dx

[Multiplying and dividing by 2]

= x tan^-1x – (1/2) ∫2x/(1 + x²) dx

{Using formula ∫f'(x)/f(x) dx = ln |f(x)| + C}

= x tan^-1x – (1/2) ln |1 + x²| + C

Hence, the integral of tan inverse x is x tan^-1x – (1/2) ln |1 + x²| + C.

Example 5: Calculate the integral of sec x with respect to x.

Solution:

Firstly, we multiply and divide the integrand with (sec x + tan x).

∫ sec x dx = ∫ sec x · (sec x + tan x) / (sec x + tan x) dx

= ∫ (sec²x + sec x tan x) / (sec x + tan x) dx

Now assume that sec x + tan x = t.

Differentiating with respect to x.

Then (sec x tan x + sec²x) dx = dt.

Substituting these values in the above integral,

∫ sec x dx = ∫ dt / t = ln |t| + C

Substituting t = sec x + tan x back here,

Hence, ∫ sec x dx = ln |sec x + tan x| + C.

Integral of tan x is ln |sec x| + C. Integral of tan x refers to finding the integration of the trigonometric function tan x with respect to x which can be mathematically formulated as ∫tan x dx. The tangent function, tan x, is an integrable trigonometric function that is defined as the ratio of the sine and cosine functions.

This article discusses the formula for the integral of tan x along with derivation, definite integral of tan x, and integral of tan inverse x. We will also discuss some solved examples based on the integral of tan x along with Practice Questions and FAQs.