Problem 57 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Determine the antiderivative of the following function: $$\int e^{x+2} dx$$ Answer: The antiderivative of the given function is $$e^{x+2} + C$$, where \(C\) is the constant of integration.

Step 1: Apply the substitution rule

Let $$u=x+2$$ so then, $$\frac{du}{dx}=1$$ and $$dx=du$$. Now we can substitute \(u\) into the integral function. $$\int e^{x+2} dx=\int e^{u} du$$

Step 2: Evaluate the integral

Using the basic integral formula for the exponential function, we get: $$\int e^{u} du = e^{u} + C$$ where \(C\) is the constant of integration.

Step 3: Substitute back into the original variable

Since we had substituted \(u=x+2\) in step 1, now we replace \(u\) with the original variable \(x\) to get the result in terms of \(x\). $$e^{u}+C=e^{x+2}+C$$

Step 4: Check the solution by differentiation

Now we must verify our solution by differentiating the result and comparing it with the original integrand function. Differentiate the antiderivative function found in step 3: $$\frac{d}{dx}(e^{x+2}+C)=\frac{d}{dx}(e^{x+2})+\frac{d}{dx}(C)$$ Using the properties of derivative and the derivative of the exponential function, we get: $$e^{x+2}+0=e^{x+2}$$ Since this matches the original integrand function, our solution is correct.