Solving the Integral of csc²x Using Trigonometric Identities and Integration Techniques

integral of csc²x

The integral of csc²x can be solved by using trigonometric identities and integration techniques

The integral of csc²x can be solved by using trigonometric identities and integration techniques.

To start, let’s recall the trigonometric identity for the square of cosecant:
csc²x = 1 + cot²x
This identity will be useful in simplifying the integrand.

Now, we can rewrite the integral as follows:
∫ csc²x dx

Using the trigonometric identity, we have:
∫ (1 + cot²x) dx

Breaking this down further, we get:
∫ dx + ∫ cot²x dx

The first integral, ∫ dx, is simply the antiderivative of a constant and yields x:
x + ∫ cot²x dx

Now, we need to integrate the second term, ∫ cot²x dx. This can be done using a trigonometric substitution.

Let’s substitute cot(x) = u, which implies dx = -du / (u² + 1).
This gives us:
x + ∫ -du / (u² + 1)

At this point, we have u as cot(x) and need to rewrite our integral in terms of x. For this, we use the relationship between cot(x) and x.

Recall that cot(x) = cos(x) / sin(x).
Rearranging this, we have sin(x) / cos(x) = 1 / tan(x).

Thus, we can rewrite u as:
u = cot(x) = 1 / tan(x) = cos(x) / sin(x)

Now, let’s solve for sin(x) and cos(x) using the Pythagorean identity sin²(x) + cos²(x) = 1.

sin²(x) + (cos²(x) / sin²(x)) = 1
Multiply throughout by sin²(x):
sin⁴(x) + cos²(x) = sin²(x)

Simplifying:
sin⁴(x) = sin²(x) – cos²(x)
sin⁴(x) = 1 – cos²(x) – cos²(x)
sin⁴(x) + cos²(x) = 1 – 2cos²(x)
1 – cos²(x) = 1 – (1 – sin²(x))
1 – cos²(x) = sin²(x)

Substituting sin²(x) = 1 – cos²(x) into our u expression:
u = cos(x) / sin(x) = 1 / √(1 – cos²(x))

Now we can rewrite our integral in terms of u by substituting:
x + ∫ -du / (u² + 1)
x – ∫ du / (u² + 1)

We can now integrate this new expression:
x – arctan(u) + C

Finally, substituting back u = cot(x):
x – arctan(cot(x)) + C

Recall that arctan(cot(x)) = x, so the final solution is:
∫ csc²x dx = x – x + C = C

Therefore, the integral of csc²x is equal to a constant value C.

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