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Chapter 4: Problem 57

Determine the following indefinite integrals. Check your work by differentiation. $$\int e^{x+2} d x$$

Short Answer

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Question: Find the indefinite integral of the function \(f(x) = e^{x+2}\). Answer: The indefinite integral of the function \(f(x) = e^{x+2}\) is given by \(\int e^{x+2} dx = e^{x+2} + C\), where \(C\) is the constant of integration.

Step by step solution

01

Substitution of variables

We start by letting a new variable \(u = x + 2\). Then differentiate \(u\) with respect to \(x\) to obtain \(du = dx\).
02

Rewrite the integral in terms of the new variable

Substitute \(u\) and \(du\) into the integral, so we have: $$\int e^u du$$
03

Solve the integral in terms of the new variable

Now, we can find the indefinite integral: $$\int e^u du = e^u + C$$ Where \(C\) is the constant of integration.
04

Substitute back to the original variable

Substitute \(x+2\) for \(u\) in our result to find the anti-derivative in terms of \(x\): $$e^{x+2} + C$$
05

Checking the result by differentiation

Now, let's differentiate our result to check if it matches the original function: $$\frac{d}{dx}(e^{x+2} + C) = e^{x+2}$$ As expected, the derivative of our result matches the original function. Therefore, the indefinite integral of \(e^{x+2}\) is: $$\int e^{x+2} dx = e^{x+2} + C$$

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