How to solve an integration problem involving exponential functions?

Thread starter stunner5000pt
Start date Sep 28, 2006
Tags

Integration

In summary, a simple integration problem involves finding the definite or indefinite integral of a given function. The basic steps for solving such a problem include identifying the function, determining the limits of integration, applying integration rules and techniques, and evaluating the result. Common integration rules and techniques include the power rule, integration by parts, substitution, and trigonometric substitution. Simple integration problems have various real-life applications, such as calculating areas and volumes, estimating revenue and population growth. Some mistakes to avoid when solving a simple integration problem include forgetting the constant of integration, misapplying rules, and not using appropriate limits.

Sep 28, 2006

stunner5000pt

How do i solve
[tex] \int_{0}^{\infty} \lambda^2 r e^{-\lambda r} dr [/tex]

if i were to integrate it i get
[tex] \left[-e^{-lambda r} (1 + r \lambda)\right]_{0}^{\infty}[/tex]

what is [tex] \lim_{r \rightarrow \infty} \frac{r}{e^r} [/tex]

is it zero?? by virtue of e increasing faster than r ??

Last edited: Sep 28, 2006

Physics news on Phys.org

Sep 29, 2006

dextercioby

Science Advisor

Homework Helper

Insights Author

The exponential increases faster than any polynomial.

Daniel.

FAQ: How to solve an integration problem involving exponential functions?

What is a simple integration problem?

A simple integration problem is a mathematical exercise that involves finding the definite or indefinite integral of a given function. This involves finding the area under the curve of the function between two given limits.

What are the basic steps for solving a simple integration problem?

The basic steps for solving a simple integration problem are as follows:

Identify the function to be integrated
Determine the limits of integration
Apply integration rules and techniques to find the antiderivative of the function
Plug in the limits of integration and evaluate the definite integral or leave the result as an indefinite integral

What are some common integration rules and techniques?

Some common integration rules and techniques include:

Power rule: ∫xⁿ dx = xⁿ⁺¹ / (n+1) + C
Integration by parts: ∫u dv = uv - ∫v du
Substitution: ∫f(g(x)) g'(x) dx = ∫f(u) du
Trigonometric substitution: ∫f(sin x) dx = -cos x + C

What are some real-life applications of simple integration problems?

Simple integration problems have many real-life applications, including:

Calculating the area under a curve in physics, such as finding the work done by a variable force
Determining the volume of irregularly shaped objects in engineering
Estimating the total revenue or profit in business using the concept of marginal revenue and cost
Modeling population growth in biology using the logistic growth equation

What are some common mistakes to avoid when solving a simple integration problem?

Some common mistakes to avoid when solving a simple integration problem include:

Forgetting to add the constant of integration when evaluating indefinite integrals
Misapplying integration rules or techniques
Forgetting to substitute back in the original variable when using substitution
Not checking for symmetry or using appropriate limits when calculating definite integrals