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Integral of Cosec x

To find the integral of cosec x, we will have to use trigonometry. Cosec x is the reciprocal of sin x and cot x is equal to (sin x)/(cos x). We can do the integration of cosecant x in multiple methods such as:

• By substitution method
• By partial fractions
• By trigonometric formulas

We have different formulas for integration of cosec x and let us derive each of them using each of the above-mentioned methods. Also, we will solve some examples related to the integral of cosec x.

 1 What is the Integral of Cosec x? 2 Integral of Cosec x by Substitution Method 3 Integral of Cosec x by Partial Fractions 4 Integral of Cosec x by Trigonometric Formulas 5 FAQs on Integral of Cosec x

What is the Integral of Cosec x?

The integral of cosec x is denoted by ∫ cosec x dx (or) ∫ csc x dx and its value is ln |cosec x – cot x| + C. This is also known as the antiderivative of cosec x. We have multiple formulas for this. But the more popular formula is, ∫ cosec x dx = ln |cosec x – cot x| + C. Here “ln” represents the natural logarithm and ‘C’ is the constant of integration. Multiple formulas for the integral of cosec x are listed below:

• ∫ cosec x dx = ln |cosec x – cot x| + C [OR]
• ∫ cosec x dx = – ln |cosec x + cot x| + C [OR]
• ∫ cosec x dx = (1/2) ln | (cos x – 1) / (cos x + 1) | + C [OR]
• ∫ cosec x dx = ln | tan (x/2) | + C

We use one of these formulas according to necessity.

We will prove each of these formulas in each of the mentioned methods.

Integral of Cosec x by Substitution Method

We can find the integral of cosec x proof by substitution method. For this, we multiply and divide the integrand with (cosec x – cot x). But why do we need to do this? Let us see.

∫ cosec x dx = ∫ cosec x · (cosec x – cot x) / (cosec x – cot x) dx

= ∫ (cosec2x – cosec x cot x) / (cosec x – cot x) dx

Now assume that cosec x – cot x = u.

Then (-cosec x cot x + cosec2x) dx = du.

Substituting these values in the above integral,

∫ cosec x dx = ∫ du / u = ln |u| + C

Substituting u = cosec x – cot x back here,

∫ cosec x dx = ln |cosec x – cot x| + C

Hence we proved the first formula of integral of csc x.

Let us derive another formula from this formula by using the fact cosec x – cot x = 1 / (cosec x + cot x). Then

∫ cosec x dx = ln |1/(cosec x + cot x)| + C

= ln |(cosec x + cot x)-1| + C

By a property of logarithms, ln am = m ln a. So

∫ cosec x dx = – ln |cosec x + cot x| + C

Hence we have proved the second formula of integral of csc x.

Integral of Cosec x by Partial Fractions

To find the integration of cosec x proof by partial fractions, we have to use the fact that cosec x is the reciprocal of sin x. i.e., cosec x = 1/(sin x). Wait! How can this be turned into partial fractions? Let us see.

∫ cosec x dx = ∫ 1/(sin x) dx

Multiplying and dividing this by sin x,

∫ cosec x dx = ∫ (sin x) / (sin2x) dx

Using one of the trigonometric identities,

∫ cosec x dx = ∫ (sin x dx) / (1 – cos2x) dx

Now, assume that cos x = u. Then -sin x dx = du. Then the above integral becomes

∫ cosec x dx = ∫ du / (u2 – 1)

By partial fractions, 1/(u2 – 1) = 1/2 [1/(u – 1) – 1/(u + 1)]. Then

∫ cosec x dx = (1/2) ∫ [ 1/(u – 1) – 1/(u + 1) ] du

= (1/2) [ ln |u – 1| – ln |u + 1| ] + C

(We can get this by finding the integrals ∫ [ 1/(u – 1) ] du and ∫ [ 1/(u + 1) ] du separately by substituting 1 + u = p and 1 – u = q respectively).

By a property of logarithms, ln m – ln n = ln (m/n). So

∫ cosec x dx = (1/2) ln | (u – 1) / (u + 1) | + C

Substituting u = sin x back here,

∫ cosec x dx = (1/2) ln | (cos x – 1) / (cos x + 1) | + C

Hence proved.

Integral of Cosec x by Trigonometric Formulas

We can prove that the integral of cosec x to be ln | tan (x/2) | + C by using trigonometric formulas. We can write cosec x as 1/(sin x).

$$\int \operatorname{cosec} x d x=\int \frac{1}{\sin x} d x$$

Using half-angle formulas, sin A = 2 sin A/2 cos A/2. Applying this,

$$\int \operatorname{cosec} x d x=\int \frac{1}{2 \sin \frac{x}{2} \cos \frac{x}{2}} d x$$

Multiply and dividing the denominator by cos x/2,

\begin{aligned} \int \operatorname{cosec} x d x &=\int \frac{1}{2\left(\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}\right) \cos ^{2} \frac{x}{2}} d x \\ &=\int \frac{\left(\frac{1}{2}\right) \sec ^{2} \frac{x}{2}}{\tan \frac{x}{2}} d x \end{aligned}

Now, assume that tan (x/2) = u. From this, (1/2) sec2(x/2) dx = du.

∫ cosec x dx = ∫ du/u = ln |u| + C

Substituting u = tan (x/2) back,

∫ cosec x dx = ln | tan (x/2) | + C

Hence proved the formula of integral of csc x.

Important Notes Related to Integration of Csc x:

Here are the formulas of integral of csc x with the respective methods of proving them.

• Using the substitution method,
∫ csc x dx = ln |csc x – cot x| + C (or)
∫ csc x dx = – ln |csc x + cot x| + C
• Using partial fractions,
∫ csc x dx = (1/2) ln | (cos x – 1) / (cos x + 1) | + C
• Using trigonometric formulas,
∫ csc x dx = ln | tan (x/2) | + C

Topics Related to Integral of Cosec x:

Here are some topics that you may find helpful while doing the integration of cosec x.

Solved Examples on Integration of Cosec x

1. Example 1: Evaluate the definite integral ∫₀π/2 cosec x dx if it converges.

Solution:

The integral of cosec x is,

∫ cosec x dx = ∫ cosec x dx = ln |cosec x – cot x| + C

∫₀π/2 cosec x dx is obtained by applying the limits 0 and π/2 for this. Then

∫₀π/2 cosec x dx = [ ln |cosec π/2 – cot π/2| + C ] – [ ln |cosec 0 – cot 0| + C ] = Diverges

This is because cosec 0 = ∞.

Answer: ∫₀π/2 cosec x dx diverges.

2. Example 2: What is the value of ∫ (csc x – cot x) dx?

Solution:

We know that the integrals of cosecant x and cot x are ln |csc x – cot x| and ln |sin x| respectively. Thus,

∫ (csc x – cot x) dx = ∫ csc x dx – ∫ cot x dx

= ln |csc x – cot x| – ln |sin x| + C

Answer: ∫ (csc x – cot x) dx = ln |csc x – cot x| – ln |sin x| + C.

3. Example 3: Find the value of ∫ (cosec2x + cosec x) dx.

Solution:

∫ (cosec2x + cosec x) dx = ∫ cosec2x dx + ∫ cosec x dx

We know that ∫ cosec2x dx = -cot x and ∫ cosec x dx = ln |cosec x – cot x| + C. So

∫ (cosec2x + cosec x) dx = -cot x + ln |cosec x – cot x| + C

Answer: ∫ (cosec2x + cosec x) dx = -cot x + ln |cosec x – cot x| + C.

FAQs on Integral of Cosec x

What is the Integral of Cosec x?

There are multiple formulas for the integral of cosec x. But the most popular formula that we use is ∫ cosec x dx = ln |cosec x – cot x| + C, where C is the integration constant.

How do You Find the Antiderivative of Cosec x?

The antiderivative of cosec x is mathematically writen as ∫ cosec x dx. Multiply and divide cosec x by (cosec x – cot x), we get, ∫ cosec x dx = ∫ cosec x · (cosec x – cot x) / (cosec x – cot x) dx = ∫ (cosec2x – cosec x cot x) / (cosec x – cot x) dx. By assuming cosec x – cot x = u. Then (-cosec x cot x + cosec2x) dx = du. Then the above integral becomes ∫ cosec x dx = ∫ du / u = ln |u| + C = ln |cosec x – cot x| + C.

What is the Integral of Csc x Cot x?

The derivative of csc x is -csc x cot x. From this, the derivative of -csc x is csc x cot x. Thus, ∫ csc x cot x dx = – csc x + C.

What is the Integration of Cosec^2x?

We know that the derivative of cot x is -cosec2 x. So the derivative of -cot x is cosec2x. Thus, the integral of cosec2x is -cot x. i.e., ∫ cosec2x dx = -cot x + C.

How to Find the Integral of Cosec x by Substitution?

We can write ∫ cosec x dx as ∫ cosec x dx = ∫ cosec x · (cosec x – cot x) / (cosec x – cot x) dx. Now assume cosec x – cot x = u. Differentiating, (-cosec x cot x + cosec2x)dx = du. Then, ∫ cosec x dx = ∫ du / u = ln |u| + C = ln |cosec x – cot x| + C.

What is the Integral of cosec2x cot x?

We can write ∫ cosec2x cot x dx as ∫ cosec x (cosec x cot x) dx. Assume that cosec x = u then -cosec x cot x dx = du. Then the above integral becomes, ∫ -u du = -u2/2 + C = (-cosec2x)/2 + C.

What is the Integral of Cosec x from 0 to π/4?

We know that the integral of cosecantx is ln |cosec x – cot x|. Applying the limits 0 and π/4, we get ∫₀π/4 cosec x dx = ln |cosec π/4 – cot π/4| – ln |cosec 0 – cot 0| = diverges as cosec 0 is undefined.

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