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Chapter 5: Problem 36

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\ln 8} e^{x} d x$$

Short Answer

Expert verified
Answer: The value of the definite integral is $7$.

Step by step solution

01

Identify the integrand and the limits

We are given the integral: $$\int_{0}^{\ln 8} e^{x} dx$$ Here, the integrand function is \(e^x\). The lower limit and the upper limits of the integral are \(0\) and \(\ln 8\), respectively.
02

Find the antiderivative of the integrand function

To find the antiderivative of \(e^x\), we need to find a function whose derivative is \(e^x\). The derivative of \(e^x\) is itself, so we have: $$F(x) = e^x$$ Where \(F(x)\) is the antiderivative of \(e^x\).
03

Apply the Fundamental Theorem of Calculus to evaluate the definite integral

Now that we have the antiderivative, we can use the Fundamental Theorem of Calculus: $$\int_{a}^{b} f(x) dx = F(b) - F(a)$$ In our case, \(a = 0\), \(b = \ln 8\), \(f(x) = e^x\), and \(F(x) = e^x\). Plug these values into the formula: $$\int_{0}^{\ln 8} e^x dx = F(\ln 8) - F(0)$$
04

Calculate the value of the definite integral

Now, substitute the values into the formula and calculate the value of the definite integral: $$F(\ln 8) - F(0) = e^{\ln 8} - e^0$$ Since \(e^{\ln a} = a\), we have: $$e^{\ln 8} = 8$$ And \(e^0 = 1\). Thus, the value of the definite integral is: $$\int_{0}^{\ln 8} e^x dx = 8 - 1 = 7$$ Therefore, the value of the definite integral is \(7\).

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