The integral of cosec (cosecant) is -ln|cosec(x) + cot(x)| + C, where C is the constant of integration.
Understanding the Integral
- Cosecant (cosec): The cosecant function is the reciprocal of the sine function. It is denoted as cosec(x) or csc(x).
- Integral: The integral represents the area under the curve of a function.
- Constant of Integration (C): The constant of integration is added to account for the fact that the derivative of a constant is always zero.
Practical Applications
The integral of cosecant is used in various fields, including:
- Physics: Calculating the motion of objects under the influence of gravity.
- Engineering: Analyzing the behavior of waves and oscillations.
- Mathematics: Solving differential equations and evaluating definite integrals.
Example
Let's find the integral of cosec(x) from x = π/4 to x = π/2:
∫(π/4)^(π/2) cosec(x) dx = [-ln|cosec(x) + cot(x)|]_(π/4)^(π/2)
= [-ln|cosec(π/2) + cot(π/2)|] - [-ln|cosec(π/4) + cot(π/4)|]
= [-ln|1 + 0|] - [-ln|√2 + 1|]
= ln(√2 + 1)Therefore, the definite integral of cosec(x) from π/4 to π/2 is ln(√2 + 1).
