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Chapter 5: Problem 52

Use a change of variables to evaluate the following definite integrals. $$\int_{0}^{\ln 4} \frac{e^{x}}{3+2 e^{x}} d x$$

Short Answer

Expert verified
Question: Evaluate the definite integral using a change of variables (substitution): $$\int_{0}^{\ln 4} \frac{e^{x}}{3+2 e^{x}} d x$$ Answer: $$\int_{0}^{\ln 4} \frac{e^{x}}{3+2 e^{x}} d x = \frac{1}{2} \ln\left(\frac{11}{5}\right)$$

Step by step solution

01

Choose a substitution variable

Let us choose the substitution variable \(u\) as follows: $$u = 3 + 2e^x$$ Now, we need to find the differential \(du\) and the new limits of integration.
02

Find the differential and new limits of integration

First, find the derivative of \(u\) with respect to \(x\) and then multiply both sides by \(dx\) to find the differential \(du\): $$\frac{du}{dx} = 2e^x \Rightarrow du = 2e^x dx$$ Now, find the new limits of integration by substituting the original limits \(x=0\) and \(x=\ln4\). For \(x=0\): $$u = 3 + 2e^0 = 3 + 2 = 5$$ For \(x=\ln4\): $$u = 3 + 2e^{\ln4} = 3 + 2(4) = 11$$ The new integral will be with respect to \(u\), with limits of integration between \(5\) and \(11\).
03

Rewrite the integral using the substitution

Now we will rewrite the original integral using our substitution: $$\int_{0}^{\ln 4} \frac{e^{x}}{3+2 e^{x}} dx = \int_{5}^{11} \frac{1}{2} \cdot \frac{1}{u} du$$ Notice that \(\frac{1}{2}\) is a constant, so we can take it out of the integral. $$\frac{1}{2} \int_{5}^{11} \frac{1}{u} du$$
04

Evaluate the integral

Now we have a simple integral of the form: $$\int \frac{1}{u} du = \ln|u| + C$$ Using the new limits of integration from step2, we can apply the Fundamental Theorem of Calculus and then factor out the constant: $$\frac{1}{2} \left(\ln|11| - \ln|5|\right) = \frac{1}{2} \ln\left(\frac{11}{5}\right)$$
05

Final answer

The final answer for the given integral using the change of variables is: $$\int_{0}^{\ln 4} \frac{e^{x}}{3+2 e^{x}} d x = \frac{1}{2} \ln\left(\frac{11}{5}\right)$$

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