Integration of an exponential function

In summary, an exponential function is a type of growth function that increases or decreases at a rate proportional to its current value. Integration of an exponential function involves finding the antiderivative or indefinite integral of the function, which helps in calculating the area under the curve and is important in various fields of science and engineering. The general formula for integrating an exponential function is ∫a^x dx = (a^x)/ln(a) + C, and to solve definite integration, one must evaluate the indefinite integral at the upper and lower limits of integration and subtract the lower limit value from the upper limit value.
  • #1
Prof Sabi
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Member warned that the homework template must not be deleted
How to Integrate it:::

∫e^(ax²+bx+c)dx

Or in general e raised to quadratic or any polynomial. I am trying hard to recall but I couldn't recall this integration. I tried using By-parts but the integration goes on and on.
 
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  • #2
I am posting for first time in HW section If I did any mistake please please don't remove my post but edit it the way you want Mod Uncle. Thank youuu :D
 
  • #3
That can't be solved in general with elementary functions. It requires the error function. https://en.wikipedia.org/wiki/Error_function
The imaginary error function is
##\int e^{x^2} dx = \frac{\sqrt{\pi}}{2} \mathrm{erfi}(x)##

If you complete the square
##ax^2+bx+c=a(x+d)^2+f##
##d=\frac{b}{2a}##
##f=c-\frac{b^2}{4a}##
Then you get
##\int e^{a(x+d)^2+f} dx=\int e^{a(x+d)^2} e^f dx##
Then you can do some u substitution to put it into the form of the error function
 
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  • #4
Khashishi said:
That can't be solved in general with elementary functions. It requires the error function. https://en.wikipedia.org/wiki/Error_function
The imaginary error function is
##\int e^{x^2} dx = \frac{\sqrt{\pi}}{2} \mathrm{erfi}(x)##

If you complete the square
##ax^2+bx+c=a(x+d)^2+f##
##d=\frac{b}{2a}##
##f=c-\frac{b^2}{4a}##
Then you get
##\int e^{a(x+d)^2+f} dx=\int e^{a(x+d)^2} e^f dx##
Then you can do some u substitution to put it into the form of the error function
:eek: Umm... Looks like probably I haven't learn this function, by the way I'm in 12th year of High School..
 
  • #5
Prof Sabi said:
How to Integrate it:::

∫e^(ax²+bx+c)dx

Or in general e raised to quadratic or any polynomial. I am trying hard to recall but I couldn't recall this integration. I tried using By-parts but the integration goes on and on.

To add to what Khashishi said in post #3, it is PROVABLE that no elementary antiderivative exists for ##a \neq 0## in your integral; that is, no finite expression can possibly exist for the antiderivative in the standard functions--powers, roots, trig functions, exponentials and the inverses of all these. Of course, there are non-finite expressions---such as infinite series and the like---that give the antiderivative, but no finite formula. Even if you allow yourself to write a formula 10 million pages in length, you still could not do it!

Let me emphasize: that result is a rigorously-proven fact. No matter how smart you are or how long you search, you can never find what you are looking for.

Google "non-elementary integration"; for example, see
http://www.sosmath.com/calculus/integration/fant/fant.html
for the basic facts and
https://www.math.dartmouth.edu/~dana/bookspapers/elementary.pdf
for some proofs.
 
Last edited:
  • #6
Thread moved. Please post questions involving integration or differentiation in the Calculus & Beyond section, not the Precalc section.
 

1. What is an exponential function?

An exponential function is a mathematical function in the form of f(x) = a^x, where a is a constant and x is a variable. It is a type of growth function that increases or decreases at a rate proportional to its current value.

2. What is integration of an exponential function?

Integration of an exponential function involves finding the antiderivative or indefinite integral of the function. It is the reverse process of differentiation, and it helps in calculating the area under the curve of the exponential function.

3. Why is integration of an exponential function important?

Integration of an exponential function is important in various fields of science and engineering, such as physics, biology, and economics. It helps in solving real-world problems involving growth and decay, and in modeling natural phenomena.

4. What is the general formula for integrating an exponential function?

The general formula for integrating an exponential function is ∫a^x dx = (a^x)/ln(a) + C, where C is the constant of integration.

5. How do you solve definite integration of an exponential function?

To solve definite integration of an exponential function, evaluate the indefinite integral at the upper and lower limits of integration, and then subtract the lower limit value from the upper limit value. The result will be the definite integral of the exponential function within the given limits.

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