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How to integrate a Natural Exponential Function by an easy method

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zadocksmath | Aug 31, 2009 |

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Many problem Solvers of Integral Calculus often use a long method to solve integration problems whose integrand is a Natural Exponential Function.This article will show a short method to evaluate such an integral.

Things You Will Need

Paper, pencil, and a creative mind.

Step 1

We are going to use an example problem to Illustrate the short method of evaluating an Integral whose Integrand is a Natural exponential function. To see the problem/Integral, Please click on the Image.

Step 2

3x2 is the Exponent of e, and the Derivative of 3x2 is 6x. We will look to see if the Derivative of the Exponent of the Base e is being multiplied by the Natural Exponential Function. Please click on the Image for a better understanding.

Step 3

Let us look at the Integral whose Integrand is a Natural Exponential Function. The derivative of the Exponent, which is supposed to Multiply the Natural Exponential Function, differs by a Constant Multiple. Please click on the Image for a better understanding.

Step 4

We will Multiply that Constant by the Reciprocal of the Constant of the Derivative of the Exponent, and write the Answer/Solution to the Integral as the Natural Exponential Function multiplied by the product of the constant plus a constant C. Please click on the Image for a better understanding.
There are two basic types of U-Substitutions that can be used when we are working with integrals. The first is U-Substitution done mentally, and the other is writing out the steps in full detail. In most assesment tests and entrance exams, the calculus problems will allow mental U-substitutions to be applied. This article has provided an example of one of those problems.

Tips & Warnings

Do not assume that this method will work for all Integration problems of this form. It is the reader's responsibility to understand when the mental U-Substitution method can be applied.