How to Find the Integral of e^x+x^e

$$\int e^x + x^e \ dx$$

Finding the integral of \(\mathbf{f(x)=e^x+x^e}\) can be tricky at first glance. I’m sure you’re probably familiar with how to take the integral of the \(\mathbf{e^x}\) part of it. In general, you’ll just want to remember that $$\int e^x \ dx = e^x$$

However, the \(\mathbf{x^e}\) piece looks a little weird. At first glance it may even look completely foreign. What have you seen that looks like this piece of the function?

Let’s think about what’s going on here

If we look back at our original integral, you’ll want to notice the dx at the end of it. Remember we had $$\int e^x + x^e \ dx$$

The dx is important because it indicates what letter is our variable that we will be integrating with respect to. Since it’s dx, that means we will be integrating with respect to x. Therefore, e will need to be treated as a constant.

In fact, e is a known constant with a specific value. It’s kind of like \(\mathbf{\pi}\). We know that \(\mathbf{e \approx 2.71}\). So when we are trying to integrate the \(\mathbf{x^e}\) part of this function, what we are integrating just comes down to “a variable raised up to a constant power.”

When you are trying to integrate x raised up to some constant power, you would want to use the power rule. The power rule for integrating tells us that if n is some constant other than n = -1, then $$\int{x^n} \ dx \ = \ \frac{x^{n+1}}{n+1} \ = \ \frac{1}{n+1} \cdot x^{n+1}$$

So since e is a constant, we can basically just replace the n in the above formula with e to find the integral of \(\mathbf{x^e}\). Using this, we can see that $$\int{x^e} \ dx \ = \ \frac{x^{e+1}}{e+1} \ = \ \frac{1}{e+1} \cdot x^{e+1}$$

Putting these integrals together

Now that we have figured out how to integrate the \(\mathbf{x^e}\) part of our function, let’s go back to the original integral. As we do this, let’s also think about one of the basic integral properties. Specifically, the one that tells us what to do about integrating a function that is a sum of two simpler functions. $$\int f(x)+g(x) \ dx \ = \int f(x) \ dx + \int g(x) \ dx$$

So as a result of this, we can break down our problem into two simpler integrals that we already know. $$\int e^x + x^e \ dx \ = \int e^x \ dx + \int x^e \ dx$$

And since we already found both of these integrals earlier, we know $$\int e^x + x^e \ dx \ = \ e^x + \frac{x^{e+1}}{e+1} + C$$

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