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.
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.
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.
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.
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.