Evaluate the following integral: [tex]$\int 5 e^{3 x} d x$[/tex]

Apply the constant multiple rule: ∫ 5 e 3 x d x = 5 ∫ e 3 x d x .
Use substitution: let u = 3 x , so d x = 3 1 ​ d u .
Rewrite and integrate: 5 ∫ e 3 x d x = 3 5 ​ ∫ e u d u = 3 5 ​ e u + C .
Substitute back: 3 5 ​ e 3 x + C .

Explanation

Problem Analysis We are asked to evaluate the indefinite integral ∫ 5 e 3 x d x .

Apply Constant Multiple Rule We can use the constant multiple rule, which states that ∫ c f ( x ) d x = c ∫ f ( x ) d x , where c is a constant. In our case, c = 5 , so we have:

Rewriting the Integral ∫ 5 e 3 x d x = 5 ∫ e 3 x d x

Apply Substitution Method Now, we can use the substitution method. Let u = 3 x , then the derivative of u with respect to x is d u / d x = 3 , so d u = 3 d x . Therefore, d x = 3 1 ​ d u .

Substituting u and dx Substituting u and d x into the integral, we get:

Rewriting the Integral with u 5 ∫ e 3 x d x = 5 ∫ e u 3 1 ​ d u = 3 5 ​ ∫ e u d u

Evaluating the Integral The integral of e u with respect to u is simply e u . So, we have:

Integrating e^u 3 5 ​ ∫ e u d u = 3 5 ​ e u + C

Substituting Back for x Finally, we substitute back u = 3 x to get the result in terms of x :

Final Result 3 5 ​ e u + C = 3 5 ​ e 3 x + C


Examples
Imagine you're modeling the growth of a bacteria population where the growth rate is proportional to the current population size. The integral of an exponential function, like the one we just solved, can help you determine the population size at any given time. Understanding how to solve these integrals allows you to predict and manage population growth in various scenarios, from microbiology to environmental science.

Answered by GinnyAnswer | 2025-07-08