∫csc(x)dx
To solve the integral ∫csc(x)dx, we can use several steps and techniques
To solve the integral ∫csc(x)dx, we can use several steps and techniques. First, let’s rewrite the integral using trigonometric identities.
Recall that csc(x) is the reciprocal of sin(x), so we can rewrite the integral as:
∫(1/sin(x))dx
Next, we can apply a technique called u-substitution, where we substitute a variable within the integral to simplify the expression. Let’s substitute u = sin(x). Then, du = cos(x)dx.
Now, we can rewrite the integral in terms of u:
∫(1/u)du
This integral is much simpler to evaluate. Using the power rule of integration, we know that the integral of 1/u is ln|u| + C, where C is the constant of integration.
Therefore, the solution to ∫csc(x)dx is ln|sin(x)| + C.
It is important to note that the absolute value | | is included because the natural logarithm function has a domain restriction where the argument must be positive. However, since sin(x) can be both positive and negative in different quadrants, we need to consider the absolute value to ensure the final expression is valid for all x values.
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