Integration is a mathematical process that involves finding the area under a curve or the accumulation of a function. It is the inverse operation of differentiation and is used to solve problems in physics, engineering, economics, and other fields.
The exponential function e^x is a fundamental mathematical function that appears in many applications. It is commonly used in integration because it has a simple derivative, making it easier to evaluate integrals involving e^x.
To evaluate e^x, you can use the power rule for integrals, which states that the integral of e^x is equal to e^x + C, where C is a constant. This means that when integrating e^x, you simply add one to the exponent and divide by the new exponent. For example, the integral of e^x^2 would be e^x^2 + C.
Yes, e^x can also be integrated using the substitution method, where you substitute u = x in the integral. This transforms the integral into the form of the natural logarithm, which can then be evaluated using the logarithm rules.
Evaluating e^x is commonly used in problems involving growth and decay, such as population growth, radioactive decay, and compound interest. It is also used in solving differential equations, which are used to model various phenomena in science and engineering.