The integral of cosec x with respect to x is -ln | cosec x + cot x | + C, where C is the constant of integration.
The integral of cosec(x) with respect to x, denoted as ∫cosec(x) dx, is found using the integration rules for trigonometric functions. The result is:
∫ cosec(x) dx = −ln ∣ cosec x + cot x ∣+C
where ln represents the natural logarithm, cosec is the cosecant function, cot is the cotangent function, and C is the constant of integration. This expression signifies that the antiderivative of cosec(x) involves a logarithmic term. The presence of the absolute value ensures that the expression is valid for all x within its domain.
This expression reveals the integral of cosec(x) as a combination of a term involving x and a logarithmic term. The logarithmic function arises due to the integral involving the cotangent function, showcasing the intricate connections between trigonometry and logarithmic functions in calculus.