Solved Examples on Integral of Cot x
Example 1: Find ∫cot 6x dx
Solution:
We have ∫cot 6x dx ——(1)
Let t = 6x
Differentiating w.r.t t
dt = 6 dx
⇒ dx = dt / 6
Putting in (1)
∫cot 6x dx = ∫cot t (dt / 6)
⇒ ∫cot 6x dx = (1 / 6) ∫cot t dt
⇒ ∫cot 6x dx = (1 / 6) [ln |sin t| + C]
⇒ ∫cot 6x dx = (1 / 6) [ln |sin (6x) | + C]
Example 2: Evaluate: ∫cot x cosec2x dx
Solution:
Let I = ∫cot x cosec2x dx —–(1)
Take t = cot x
Differentiating w.r.t t
dt = – cosec2x dx
putting in (1)
I = -∫t dt
⇒ I = -t2 / 2 + C (putting values)
⇒ I = – cot2x / 2 + C
⇒ ∫cot x cosec2x dx = – cot2x / 2 + C
Example 3: Solve ∫cot x. sec x dx
Solution:
I = ∫cot x. sec x dx
We know that,
cot x = cos x / sin x and sec x = 1 / cos x
Putting in I
I = ∫ [cos x / sin x]. [1 / cos x] dx
⇒ I = ∫ [1 / sin x] dx
⇒ I = ∫ cosec x dx
⇒ I = – ln | cosec x + cot x| + C
Example 4: Evaluate ∫cot2 x dx
Solution:
I = ∫cot2 x dx
We know that,
[d / dx] (cosec x) = – cot2 x
cot2 x = – [d / dx] (cosec x)
Putting in I
I = ∫ – [d / dx] (cosec x) dx
By the property ∫[d / dx] f(x) dx = f(x) + C
I = – cosec x + C
Integral of Cot x
Integral of Cot x is ln |sin x| + C. Cot x is among one of the trigonometric functions that is the ratio of cosine and sine. The integral of cot x is mathematically represented as ∫cot x dx = ln |sinx| + C.
In this article, we will explore the integral of cot x, the integral of cot x formula, the derivation of the integral of cot x, definite integral of cot x along with some examples based on the integral of cot x.
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