Question - Finding the Integral of an Exponential Function
Solution:
\int 7^x dx = \int e^{x \ln(7)} dx
Let u = x \ln(7) \Rightarrow du = \ln(7) dx
dx = \frac{du}{\ln(7)}
\int e^{x \ln(7)} dx = \int e^u \frac{du}{\ln(7)}
\frac{1}{\ln(7)}\int e^u du = \frac{1}{\ln(7)} e^u + C
\frac{1}{\ln(7)} e^{x \ln(7)} + C = \frac{7^x}{\ln(7)} + C
So, the integral of f(x) = 7^x is \frac{7^x}{\ln(7)} + C
Tutor Solution:
The function given is \( f(x) = 7^x \). To integrate this function with respect to \( x \), we will be using the integral formula for exponential functions:
\[ \int a^x \, dx = \frac{a^x}{\ln(a)} + C \]where \( a \) is a constant, and \( C \) is the constant of integration which appears because we are performing an indefinite integral. This formula is derived from the fact that the derivative of \( a^x \) with respect to \( x \) is \( a^x \ln(a) \), and thus we are essentially reversing that process.
In our case, \( a = 7 \), so we can write:
\[ \int 7^x \, dx = \frac{7^x}{\ln(7)} + C \]This is the integral of the function \( f(x) = 7^x \). The term \( \ln(7) \) is the natural logarithm of 7, which is a constant value. The \( + C \) at the end of the integral expression represents the constant of integration and is critical whenever we perform an indefinite integral.
So, the final answer is:
\[ \int f(x) \, dx = \int 7^x \, dx = \frac{7^x}{\ln(7)} + C \]That's the complete integral for the function provided. Remember that the base of the natural logarithm in the denominator corresponds to the base of the exponential function we are integrating.
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