Integral of x^e - Solving the Problem

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In summary, the integral from 0 to 1 of (x^e + e^x) dx can be solved by using the power rule for x^e and the fact that the integral of e^x is e^x. This method can be applied since e is just a constant with a finite value.
  • #1
tommyninetwo
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I'm practicing integrals right now and came up on a question I have not seen before nor can I find online.

Integral from 0 to 1 of (x^e + e^x) dx

I'm stumped on x^e.
 
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  • #2
In the case of [itex]x^{e}[/itex], e is just a constant. The power rule can be used.
 
  • #3
[tex]\int x^e dx= \frac{1}{e+ 1}x^{e+ 1}+ C[/tex]
 
  • #4
[itex] \int_0^1 (x^e+e^x)dx [/itex]

Note that 'e' is just a constant. It has a finite value, right?
So integrating it is just like how you integrate [itex]x^2[/itex].

The integral of [itex]e^x=e^x[/itex]. And you're done.
 

1. What is the integral of x^e?

The integral of x^e is equal to (x^(e+1))/(e+1) + C, where C is the constant of integration. This can be solved using the power rule of integration.

2. How do you solve the problem of finding the integral of x^e?

The problem can be solved by using the power rule of integration, which states that the integral of x^n is equal to (x^(n+1))/(n+1) + C. In the case of x^e, this becomes (x^(e+1))/(e+1) + C.

3. Can the integral of x^e be solved using other integration techniques?

Yes, the integral of x^e can also be solved using substitution or integration by parts. These techniques may be more complicated than using the power rule, but can also provide a solution.

4. What is the significance of the constant of integration in the integral of x^e?

The constant of integration represents the family of solutions to the integral of x^e. It is added to the solution to account for all possible values of the integral. In other words, it allows for flexibility in the solution.

5. Can the integral of x^e be solved using software or calculators?

Yes, most mathematical software and calculators have the capability to solve integrals, including the integral of x^e. However, it is important to understand the steps and techniques involved in solving the integral in order to ensure the accuracy of the solution.

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