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Chapter 5: Problem 48

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{1} 10 e^{2 x} d x$$

Short Answer

Expert verified
Question: Evaluate the definite integral: \(\int_{0}^{1} 10 e^{2 x} d x\) Answer: \(5 (e^2 - 1)\)

Step by step solution

01

Find the antiderivative

First, we need to determine the antiderivative of \(10 e^{2 x}\). To do this, we will apply basic integration techniques. Let's find the integral: $$\int 10 e^{2 x} d x$$ Since integration is a linear operation, we can factor out the constant \(10\). $$10 \int e^{2 x} d x$$ Now, let's focus on integrating \(e^{2x}\). We can use substitution method: let \(u=2x\). Then, \(du=2 dx\). Therefore, \(\frac{1}{2} du=dx\). Substituting these expressions, we get: $$10 \int e^u \frac{1}{2} du$$ Factor out the \(\frac{1}{2}\) outside the integral: $$5 \int e^u du$$ Now we can simply integrate \(e^u\) with respect to \(u\): $$5 e^u + C = 5 e^{2x} + C$$
02

Apply the Fundamental Theorem of Calculus

Now that we have the antiderivative, we can apply the Fundamental Theorem of Calculus to evaluate the definite integral: $$\int_{0}^{1} 10 e^{2 x} d x = 5 e^{2x} \Big|_0^1$$ Evaluate the antiderivative at the upper limit of integration (1) and the lower limit (0): $$5 (e^{2(1)} - e^{2(0)}) = 5 (e^2 - e^0)$$ Remember that \(e^0 = 1\). So, the final answer is: $$5 (e^2 - 1)$$

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