Understanding and Solving the Integral of csc(x) using Substitution

integral of cscx

The integral of csc(x) is one of the standard integrals in calculus

The integral of csc(x) is one of the standard integrals in calculus. To compute the integral, we need to use a technique called integration by substitution.

To begin, let’s represent csc(x) in terms of sine and cosine:

csc(x) = 1/sin(x)

Now we can rewrite the integral as:

∫(1/sin(x)) dx

Next, we choose u = sin(x) as the substitution, and compute the derivative of u with respect to x:

du/dx = cos(x)

To replace dx in the integral, we rearrange the equation du/dx = cos(x) to dx = du/cos(x). Now, we can substitute these values into the integral:

∫(1/sin(x)) dx = ∫(1/u) (du/cos(x))

Notice that sin(x) = u, so cos(x) = √(1 – sin^2(x)) = √(1 – u^2). Substituting these values into the integral, we get:

∫(1/u) (du/cos(x)) = ∫(1/u) (du/√(1 – u^2))

Now, let’s solve this integral. One way to do this is by using the natural logarithm function. We rewrite the integral as:

∫(1/u) (du/√(1 – u^2)) = ∫(1/u) (1/√(1 – u^2)) du

We can now integrate:

∫(1/u) (1/√(1 – u^2)) du = ln|u| + C

Finally, substituting back u = sin(x), we have:

∫(1/sin(x)) dx = ln|sin(x)| + C

where C represents the constant of integration.

Therefore, the integral of csc(x) is ln|sin(x)| + C.

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