integral of csc^2
The integral of csc^2x, also written as ∫csc^2x dx, requires a trigonometric identity and a substitution to solve
The integral of csc^2x, also written as ∫csc^2x dx, requires a trigonometric identity and a substitution to solve.
Let’s start by recalling the trigonometric identity for csc^2x:
csc^2x = 1 + cot^2x
Now, let’s rewrite the integral using this identity:
∫csc^2x dx = ∫(1 + cot^2x) dx
We can split this integral into two parts:
∫1 dx + ∫cot^2x dx
The integral of 1 dx is simply x + C, where C represents the constant of integration.
For the integral of cot^2x dx, we need to make a substitution. Let’s use u = cotx.
To differentiate u with respect to x, we apply the chain rule: du/dx = -csc^2x.
Rearranging this equation, we get dx = -du/csc^2x.
Now, let’s substitute these values into the integral:
∫cot^2x dx = ∫(cot^2x)(-du/csc^2x)
Simplifying, we have:
∫cot^2x dx = -∫du
Since the integral of du is simply u + C, we can write:
∫cot^2x dx = -u + C
Substituting back u = cotx:
∫cot^2x dx = -cotx + C
Finally, integrating the two parts of the original integral:
∫csc^2x dx = x – cotx + C
Therefore, the integral of csc^2x is x – cotx + C, with C representing the constant of integration.
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