Calculate the integral ∫ sec(x)tan(x) dx.

Calculate the integral ∫ sec(x)tan(x) dx.

• A. tan(x) + C
• B. sec(x) + C
• C. sec^2(x) + C
• D. ln|sec(x) + tan(x)| + C

Answer: (B)

To find the integral of sec(x)tan(x), we first need to recognize the relationship between these two functions and their derivatives. The derivative of sec(x) is sec(x)tan(x). This means that when we integrate sec(x)tan(x), we are effectively reversing the differentiation of sec(x). Thus, the integral can be expressed as: ∫ sec(x)tan(x) dx = sec(x) + C where C represents the constant of integration. This is why the answer you are looking for corresponds to sec(x) + C. Other response options do not give the correct result for this integral. For instance, tan(x) + C is not correct since the derivative of tan(x) is not sec(x)tan(x) but rather sec^2(x). Similarly, sec^2(x) + C would represent the integral of tan(x), and the option involving ln|sec(x) + tan(x)| + C corresponds to the integral of sec(x) itself, rather than sec(x)tan(x). Therefore, the integral of sec(x)tan(x) is accurately represented by sec(x) + C.

To find the integral of sec(x)tan(x), we first need to recognize the relationship between these two functions and their derivatives. The derivative of sec(x) is sec(x)tan(x). This means that when we integrate sec(x)tan(x), we are effectively reversing the differentiation of sec(x).

Thus, the integral can be expressed as:

∫ sec(x)tan(x) dx = sec(x) + C

where C represents the constant of integration. This is why the answer you are looking for corresponds to sec(x) + C.

Other response options do not give the correct result for this integral. For instance, tan(x) + C is not correct since the derivative of tan(x) is not sec(x)tan(x) but rather sec^2(x). Similarly, sec^2(x) + C would represent the integral of tan(x), and the option involving ln|sec(x) + tan(x)| + C corresponds to the integral of sec(x) itself, rather than sec(x)tan(x). Therefore, the integral of sec(x)tan(x) is accurately represented by sec(x) + C.