How to Calculate the Integral of csc(x) using Trigonometric Substitution

∫(cscx)dx

To calculate the integral of csc(x), also known as the cosecant function, we’ll need to use some trigonometric identities

To calculate the integral of csc(x), also known as the cosecant function, we’ll need to use some trigonometric identities. The integral of csc(x) can be evaluated by rewriting it in terms of sine and cosine functions.

The cosecant function, csc(x), is defined as the reciprocal of the sine function. Therefore, we can rewrite it as:

csc(x) = 1/sin(x)

Now, let’s proceed with the integral:

∫(cscx)dx = ∫(1/sin(x))dx

To evaluate this integral, we’ll use a trigonometric substitution. Let’s substitute u = cos(x), which means du = -sin(x)dx:

∫(1/sin(x))dx = ∫(1/u)(-1/sqrt(1-u^2))du

Now, we have expressed the integral in terms of the new variable u. To simplify this further, we’ll represent sqrt(1 – u^2) as sin(x):

∫(1/u)(-1/sqrt(1-u^2))du = ∫(-1/u^2)du

We can integrate this expression as it is a simple power rule:

∫(-1/u^2)du = -1/u + C

Now, we need to undo our substitution by substituting back u = cos(x) into the result:

-1/u + C = -1/cos(x) + C

Therefore, the final result of the integral is:

∫(cscx)dx = -1/cos(x) + C

Note that C represents the constant of integration and could be any arbitrary constant.

In summary, the integral of csc(x) is -1/cos(x) + C, where C is the constant of integration.

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