What is the integral of cosec^2(x) dx?

What is the integral of cosec^2(x) dx?

• A. -cot(x) + C
• B. sec(x) + C
• C. csc(x) + C
• D. cot(x) + C

Answer: (A)

The integral of cosec^2(x) can be understood through the relationship between cosecant and cotangent functions in calculus. The derivative of cotangent, which is cot(x), is known to be -cosec^2(x). Therefore, when you integrate cosec^2(x), you are essentially reversing that differentiation process. When you perform the integral, you are looking for a function whose derivative is cosec^2(x). Since the derivative of -cot(x) equals cosec^2(x), it becomes clear that the integral of cosec^2(x) is -cot(x) plus a constant of integration, C. This aligns with the fundamental theorem of calculus, which connects differentiation and integration. Thus, the expression -cot(x) + C accurately represents the integral of cosec^2(x), making it the correct answer. The other options relate to different functions whose derivatives do not match cosec^2(x), leading to their disqualification as possible answers.

The integral of cosec^2(x) can be understood through the relationship between cosecant and cotangent functions in calculus. The derivative of cotangent, which is cot(x), is known to be -cosec^2(x). Therefore, when you integrate cosec^2(x), you are essentially reversing that differentiation process.

When you perform the integral, you are looking for a function whose derivative is cosec^2(x). Since the derivative of -cot(x) equals cosec^2(x), it becomes clear that the integral of cosec^2(x) is -cot(x) plus a constant of integration, C. This aligns with the fundamental theorem of calculus, which connects differentiation and integration.

Thus, the expression -cot(x) + C accurately represents the integral of cosec^2(x), making it the correct answer. The other options relate to different functions whose derivatives do not match cosec^2(x), leading to their disqualification as possible answers.