Final answer:
The integral of sec(kx)tan(kx) is ln|sec(kx)| + C, which corresponds to Option 2 among the presented options. This results from the relationship that sec(kx)tan(kx) is the derivative of sec(kx) multiplied by the derivative of the inner function due to the chain rule in differentiation.
Step-by-step explanation:
sec(kx)tan(kx) is ln|sec(kx)| + C, The integral of sec(kx)tan(kx) is a common integral in calculus, especially when dealing with trigonometric functions. To find this integral, we need to recognize a pattern or use a trigonometric identity. In this case, the derivative of sec(kx) is sec(kx)tan(kx) times the derivative of the inner function, which in this case is k, due to the chain rule. The correct answer is Option 2: ∫sec(kx)tan(kx)dx = ln|sec(kx)| + C.
To find the integral of sec(kx)tan(kx), we can use the substitution method. Let u = sec(kx), then du = ksec(kx)tan(kx)dx. Rearranging the equation, we have dx = (1/ksec(kx))du. Substituting into the integral, we get ∫(1/u)du. This integrates to ln|u| + C. Substituting back u = sec(kx), we obtain ln|sec(kx)| + C.Therefore, the correct option is Option 2: ∫sec(kx)tan(kx)dx = ln|sec(kx)| + CSo, the integral is: OPTION 1: No, because sec(kx) is not its own derivative. OPTION 2: Yes, because the integral of sec(kx)tan(kx) is the natural logarithm of the absolute value of sec(kx) plus a constant, which is ln|sec(kx)| + C. OPTION 3: No, because the integral of tan(kx) does not involve cot(kx) OPTION 4: No, because csc(kx) is related to the cosecant function, not the secant and tangent functions.Therefore, the correct answer is Option 2: ∫sec(kx)tan(kx)dx = ln|sec(kx)| + C.