Question

What is the integral of csc(kx)cot(kx)?

Option 1: ∫csc(kx)cot(kx)dx = -csc(kx) + C
Option 2: ∫csc(kx)cot(kx)dx = -cot(kx) + C
Option 3: ∫csc(kx)cot(kx)dx = ln|csc(kx)| + C
Option 4: ∫csc(kx)cot(kx)dx = -ln|csc(kx)| + C

Answer 1

The integral of csc(kx)cot(kx) can be evaluated using a trigonometric identity. The correct answer is -cot(kx) + C.

Step-by-step explanation:

The integral of csc(kx)cot(kx) can be evaluated using a trigonometric identity. We know that csc(x) = 1/sin(x) and cot(x) = cos(x)/sin(x). So, substituting these values in the integral, we get:

∫csc(kx)cot(kx)dx = ∫1/(sin(kx)) * cos(kx)/(sin(kx)) dx

Simplifying further, we get:

∫(cos(kx))/(sin2(kx)) dx

The integral of the above expression is -cot(kx) + C, where C is the constant of integration. Therefore, option 2: ∫csc(kx)cot(kx)dx = -cot(kx) + C is the correct answer.