Formula, Proof, Examples l Integration of Cot x – Cuemath

Before going to learn what is the integral of cot x, let us recall few facts about the cot (or) cotangent function. In a right angled triangle, if x is one of the acute angles, then cot x is the ratio of the adjacent side of x to the opposite side of x. So it can be written as (cos x)/(sin x) as cos x = (adjacent)/(hypotenuse) and sin x = (opposite)/(hypotenuse). We use these facts to find the integral of cot x.

Let us learn the integral of cot x formula along with its proof and examples.

1. What is the Integral of Cot x dx? 2. Integral of Cot x Proof by Substitution 3. Definite Integral of Cot x dx 4. FAQs on Integral of Cot x

The integral of cot x dx is equal to ln |sin x| + C. It is represented by ∫ cot x dx and so ∫ cot x dx = ln |sin x| + C, where ‘C’ is the integration constant. Here,

• ‘∫’ is the symbol of integration.
• cot x is the integrand.
• dx represents that the integration is with respect to x.

We use the integration by substitution method to prove this integration of cot x formula. We will see it in the upcoming section.

Here is the derivation of the formula of integral of cot x by using integration by substitution method. For this, let us recall that cot x = cos x/sin x. Then ∫ cot x dx becomes

∫ cot x dx = ∫ (cos x)/(sin x) dx

Substitute sin x = u. Then cos x dx = du. Then the above integral becomes

= ∫(1/u) du

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

Substitute u = sin x back here,

= ln |sin x| + C

Thus, ∫ cot x dx = ln |sin x| + C.

Hence proved.

The definite integral of cot x dx is an integral with lower and upper bounds. To evaluate it, we substitute the upper and lower limits in the value of the integral cot x dx and subtract them in the same order. We will work on some definite integrals of cot x dx.

Integral of Cot x From 0 to pi/2

∫(_0^{pi/2}) cot x dx = ln |sin x| (left. right|_0^{pi/2})

= ln |sin π/2| – ln |sin 0|

= ln |1| – ln 0

We know that ln 1 = 0 and ln 0 is NOT defined. So

∫(_0^{pi/2}) cot x dx = Diverges

Therefore, the integral of cot x from 0 to π/2 diverges.

Integral of Cot x From pi/4 to pi/2

∫(_{pi/4}^{pi/2}) cot x dx = ln |sin x| (left. right|_{pi/4}^{pi/2})

= ln |sin π/2| – ln |sin π/4|

= ln |1| – ln |1/√2|

By one of the properties of logarithms, ln (m/n) = ln m – ln n. So

= ln 1 – ln 1 + ln √2

= ln √2

∫(_{pi/4}^{pi/2}) cot x dx = ln √2

Therefore, the integral of cot x from π/4 to π/2 is equal to ln √2.

Important Notes on Integral of Cot x dx:

• ∫ cot x dx = ln |sin x| + C
• Since sin x and csc x are reciprocals of each other, ∫ cot x dx = ln |csc x|-1 + C = – ln |csc x| + C
• ∫ cot2x dx = – csc x + C as d/dx(csc x) = -cot2x + C

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