csc(x)dx
To integrate csc(x)dx, we can use substitution and rewrite it in terms of an easier trigonometric function
To integrate csc(x)dx, we can use substitution and rewrite it in terms of an easier trigonometric function.
Let’s start by rewriting csc(x) in terms of sine and cosine. The reciprocal identity states that csc(x) is equal to 1/sin(x). Therefore, we have:
∫ csc(x) dx = ∫ (1/sin(x)) dx
Next, let’s use substitution to simplify the integral. We can let u = sin(x), and then find the differential du. Taking the derivative of both sides of u = sin(x), we have du = cos(x) dx.
Using this substitution, we can rewrite the integral as:
∫ (1/u) du
Now, let’s integrate ∫ (1/u) du. This can be done using logarithmic rules. The integral of 1/u is equal to ln|u| + C, where C is the constant of integration. Therefore:
∫ (1/u) du = ln|u| + C
Substituting back u = sin(x), we have:
∫ csc(x) dx = ln|sin(x)| + C
So, the integral of csc(x)dx is ln|sin(x)| + C, where C is the constant of integration.
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