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Answer the following True or False.\newline∫1sec⁡x dx=ln⁡(sec⁡x)+C\int \frac{1}{\sec x}\,dx = \ln(\sec x) + C\newlineTrue\newlineFalse

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Q. Answer the following True or False.\newline∫1sec⁡x dx=ln⁡(sec⁡x)+C\int \frac{1}{\sec x}\,dx = \ln(\sec x) + C\newlineTrue\newlineFalse
  1. Simplify Integral: Let's first simplify the integral. The secant function is the reciprocal of the cosine function, so 1/sec⁡(x)1/\sec(x) is equivalent to cos⁡(x)\cos(x). Therefore, the integral becomes:\newline∫cos⁡(x) dx\int \cos(x)\,dx
  2. Integrate cos⁡(x)\cos(x): Now we integrate cos⁡(x)\cos(x) with respect to xx. The antiderivative of cos⁡(x)\cos(x) is sin⁡(x)\sin(x), so:\newline∫cos⁡(x) dx=sin⁡(x)+C\int \cos(x)\,dx = \sin(x) + C
  3. Compare Results: We compare our result with the given statement ln⁡(sec⁡x)+C\ln(\sec x) + C. Since sin⁡(x)\sin(x) is not equivalent to ln⁡(sec⁡x)\ln(\sec x), the given statement is false.

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