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

Sau đây là tổng hợp các bài viết Cotx hữu ích nhất được tổng hợp

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

Topics Related to Integral of Cot x dx:

• Integration Formulas
• Integral of Sin x
• Integral of Sec x
• Integral Calculator
• Calculus Calculator
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