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.

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.

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 a^m = 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.

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.

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.

• Integration Formulas
• Derivative of Sec x
• Integral Calculator
• Integral of Sec x
• Derivative Formulas
• Derivative of log x
• Derivative Calculator

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 ∫₀^π/4 cosec x dx = ln |cosec π/4 - cot π/4| - ln |cosec 0 - cot 0| = diverges as cosec 0 is undefined.