How Do You Integrate ae^(a^u) with Respect to u?

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In summary, a simple integral, also known as a definite integral, is a mathematical concept used to find the area under a curve within a specific interval. To calculate a simple integral, one must determine the limits of integration, find the antiderivative of the function, and plug in the limits of integration. It differs from a double integral in that it deals with one variable and calculates area, while a double integral deals with two variables and calculates volume. The constant of integration, represented by "+ C", is added to the result of a simple integral due to the derivative of a constant always being 0. Simple integrals have various applications in fields such as physics, engineering, economics, and statistics for calculating quantities, finding average values, and
  • #1
Chaz706
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My brain is truly fried.

I know this:
[tex] \int e^u du = e^u +C [/tex]

But what do I do if I get this:
[tex] \int ae^a^u du [/tex] ? Assuming a is a non-zero constant?

EDIT: Never mind! I figured this out backwards.

[tex] \frac {d}{du} e^a^u = ae^a^u du[/tex]

Thus

[tex] \int ae^a^u du = e^a^u [/tex]
 
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  • #2
You can use substitution.
[tex]y=au\Rightarrow \frac{dy}{du}=a\Rightarrow du=\frac{dy}{a}[/tex]
Therefore,
[tex]\int ae^{au}du=\int ae^{y}\frac{1}{a}dy=\int e^{y}dy=e^{y}+C=e^{au}+C[/tex]
 
  • #3
+ C



It seems like you have already figured out the solution, but I will still provide a response for anyone else who may be struggling with this concept.

Firstly, it is important to remember that when solving an integral, we are essentially finding the antiderivative of the given function. In this case, the function is ae^a^u, where a is a non-zero constant.

To solve this integral, we can use the power rule for integration, which states that the integral of x^n is equal to (x^(n+1))/(n+1) + C. Applying this rule to our function, we get:

\int ae^a^u du = (ae^a^u)/(a+1) + C

However, this is not the final answer. We need to simplify the expression further by using the chain rule. The chain rule states that the derivative of a composite function f(g(x)) is equal to f'(g(x)) * g'(x). In our case, g(x) is a^u and f(u) is e^u. So, using the chain rule, we get:

\frac{d}{du} e^a^u = ae^a^u

Therefore, we can rewrite our integral as:

\int ae^a^u du = \frac{1}{a} * \frac{d}{du} e^a^u du = \frac{1}{a} * ae^a^u + C = e^a^u + C

And there we have it! Our final answer is e^a^u + C. I hope this helps clear up any confusion. Remember to always use the power rule and the chain rule when solving integrals. Keep practicing and it will become easier over time!
 

Related to How Do You Integrate ae^(a^u) with Respect to u?

What is a simple integral?

A simple integral, also known as a definite integral, is a mathematical concept that represents the area under a curve. It is used to find the total value of a function within a specific interval.

How is a simple integral calculated?

To calculate a simple integral, you first need to determine the limits of the integration, which are the starting and ending points of the interval. Then, you need to find the antiderivative of the function. Finally, you plug in the limits of integration into the antiderivative and subtract the result of the lower limit from the result of the upper limit.

What is the difference between a simple integral and a double integral?

A simple integral calculates the area under a curve in one dimension, while a double integral calculates the volume under a surface in two dimensions. A simple integral has one variable, while a double integral has two variables.

What is the significance of the constant of integration in a simple integral?

The constant of integration, represented by "+ C", is added to the result of a simple integral because the derivative of a constant is always 0. It represents all possible antiderivatives of the function.

What are some real-world applications of simple integrals?

Simple integrals are used in various fields such as physics, engineering, economics, and statistics to calculate quantities such as displacement, work, profit, and probability. They are also used in the process of finding the average value of a function and in solving optimization problems.

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