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Answer the following True or False.\newline∫3x dx=3x+Cln⁡3\int 3^{x}\,dx = \frac{3^{x} + C}{\ln 3}\newlineTrue\newlineFalse

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Full solution

Q. Answer the following True or False.\newline∫3x dx=3x+Cln⁡3\int 3^{x}\,dx = \frac{3^{x} + C}{\ln 3}\newlineTrue\newlineFalse
  1. Apply Exponential Function Formula: To solve the integral of 3x3^{x} with respect to xx, we need to use the formula for integrating exponential functions, which is ∫axdx=axln⁡(a)+C\int a^{x}dx = \frac{a^{x}}{\ln(a)} + C, where aa is a constant and a≠1a \neq 1.
  2. Calculate Integral of 3x3^{x}: We apply the formula to 3x3^{x}, where a=3a = 3. So, ∫3x dx=(3xln⁡(3))+C\int 3^{x}\,dx = \left(\frac{3^{x}}{\ln(3)}\right) + C.
  3. Compare Result with Given Expression: We compare the result with the given expression (3x+C)/(ln⁡3)(3^{x} + C) / (\ln 3). The integral we found, (3x/ln⁡(3))+C(3^{x} / \ln(3)) + C, is not the same as (3x+C)/(ln⁡3)(3^{x} + C) / (\ln 3) because in the given expression, both 3x3^{x} and CC are divided by ln⁡(3)\ln(3), which is incorrect.
  4. Verify Statement: Therefore, the statement ＂∫3x dx=3x+Cln⁡3\int 3^{x}\,dx = \frac{3^{x} + C}{\ln 3}＂ is False because the correct integral of 3x3^{x} is 3xln⁡(3)\frac{3^{x}}{\ln(3)} + C, not both terms divided by ln⁡(3)\ln(3).

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