Question

Consider the function F(x)=−3csc(x)cot(x). Use the given antiderivative to find the indefinite integral ∫F(x)dx

Answer 1

To find the indefinite integral of the function F(x) = -3*csc(x)*cot(x), we recognize that it is the derivative of -csc(x) multiplied by a constant 3. Thus, its antiderivative is simply -3 csc(x) plus a constant of integration, resulting in -3*csc(x) + K.

Step-by-step explanation:

To find the indefinite integral of the function F(x) = -3 · csc(x) · cot(x), we can use trigonometric identities and integration rules. The product of cosecant and cotangent can be expressed in terms of derivatives of cosecant. We know that d/dx [csc(x)] = - csc(x) · cot(x). This indicates that the given function is the negative derivative of csc(x), times a constant factor of 3.

So, the antiderivative of F(x) can be found as follows:

1. Write the function in terms of the derivative: F(x) = 3 · d/dx [-csc(x)].
2. Take the antiderivative: ∫ F(x) dx = 3 ∫ d/dx [-csc(x)] dx.
3. Recognize that the antiderivative of a derivative is the function itself (up to a constant of integration): ∫ d/dx [-csc(x)] dx = -csc(x) + C, where C is the constant of integration.
4. Apply the constant factor of 3: ∫ F(x) dx = 3(-csc(x) + C) = -3 csc(x) + 3C, or more simply, -3 csc(x) + K, where K is a new constant of integration.

Therefore, the indefinite integral of F(x) is -3 csc(x) + K, where K represents the constant of integration.