A Look at Calculus Integration

First, there are definite integrals and indefinite integrals. An indefinite integral is just the anti-derivative of a function, and is a function itself. A definite integral finds the difference between two specific values of the indefinite integral, and typically produces a numerical answer instead of a function. Definite integrals can be used to find areas and volumes of irregular figures that cannot be found with basic geometry, so long as the sides of the figure being measured follows some function that can be integrated. For example, the definite integral from 0 to 3 of x² would find the area between the x-axis and the parabola from 0 to 3. This shape is like a triangle with a curve from a parabola for a hypotenuse, and is a great example of quickly finding the area of an irregular two-dimensional shape using a definite integral.

In differential calculus, you learn that the chain rule is a key rule for taking derivatives. Its counterpart in integral calculus is the process of integration by substitution, also known as u-substitution. In general, when trying to take the integral of a function that is of the form f(g(x)) * g'(x), the result is simply f(x). However, there are a number of variations on this general theme, and it can even be extended to handle functions that have multiple variables. For a basic example, suppose you want to find the indefinite integral of (x+1)² dx. We would let u = x+1, and du = dx. After substituting u in place of x+1, and du in place of dx, we’re left with trying to take the integral of u² du, which we know from our basic patterns is just u³/3 + C. We substitute x+1 back in for u in our final answer, and quickly have (x+1)³/3 + C.

Integration in calculus is often seen a strategic process instead of a straight-forward mechanical process because of the large number of tools at your disposal for integrating functions. One very important tool is integration by parts, which is a play on the product rule for differentiation. In short, if there are two functions, call them u and v, then the integral of u dv equals uv – the integral of v du. This may seem like just another random formula, but it’s significance is that it often allows us to simplify a function that we’re taking the integral of. This strategy requires that we pick u and du in a way that the derivative of u is less complicated than u. Once we break the integral up by parts, our resulting integral contains du, but not u, meaning that the function we are taking the integral of has become simplified in the process.

How Do You Instill Impeccable Integrity at Your Home

While some values can be specific to a culture or geography, integrity is a global, universal value, like love, compassion, peace, respect and responsibility. Integrity knows no boundaries in terms of location, age, gender, class, culture, race or religion. Indeed, integrity is a human value that, when it transcends the globe, serves to make people happier, more peaceful, more aware and more thoughtful.

Integrity can seem like an abstract value or concept, but it becomes pretty clear when you start evaluating kids’ habits. According to a recent CNN report, about 85 percent of kids say that they have cheated by the time they graduate from high school, whether it is copying a neighbor’s math test, plagiarizing from the Internet, a library book or another source, texting answers during tests and more. While younger kids are more guileless and innocent about cheating and may not even realize that it’s wrong right away, it is important to address this issue – with integrity – early on, so that children grown up understanding the value of honesty and integrity.

Integrity is truly integral to an individual’s system of ethics, values and morals. As a parent, caregiver or child educator seeking to teach and show the value of integrity, this process will involve both subjective and objective measures. You can discuss the concept as a whole, but you must also share specific, real-life examples of integrity in the community and world.

Consider the following tips for sharing the global value of integrity with the children in your life:

Be a role model. With kids, actions really do speak louder than words. If you are talking to your kids about integrity and its importance, but treating store clerks with impertinence or impatience, cheating your clients out of money or talking about your friends behind their backs, the message will get more than muddled. Let your kids see you greeting strangers with a smile, letting someone know if they gave you too much change for your purchase and avoiding the temptation to cut in line. Likewise, recognize other role models, from a celebrity to a neighbor to a teacher, who your kids can identify with.

Keep your promises. This is another big deal to kids – when you say that you are going to do something, do it. Say what you mean and mean what you say. Follow through, stay true to your word and keep your promises when it comes to parenting with integrity.

Talk. Talk about shows on TV that show people acting both with and without integrity. Talk about the latest local, national and international news stories. Talk about whether your children’s friends are cheating. Talk about your day. Be open to discussing whatever is on your children’s minds when it comes to the concept of integrity.

Have consequences. If you catch your child lying, cheating, stealing or otherwise acting without integrity, don’t let the action go unnoticed or unpunished. Make it clear what the consequences for this type of behavior are and then follow through.

Business Integrity Is For Sale

How realistic is it to expect integrity from other people in the knowledge that it is almost impossible for us to maintain our own integrity? Sure, it is easy to maintain a high level of integrity as long as it doesn’t hurt or doesn’t cost money. But what do we do the moment our integrity stands in the way of a lot of money or a promotion or keeping our job? We don’t always realize it, but we do ‘stretch’ our integrity quite often.

Let me give you some examples.


Sales reps have a tendency to stretch their integrity to the max and beyond when it comes to closing deals:

“The customer is always right”. Sales people will almost never contradict a customer out of fear for losing the deal.

Customers can be rude. Very rude. I have seen sales people accepting behavior from customers that they would not accept from family or close friends. All integrity down the drain, just for money.

Every sales rep (including myself) has sold his product/service, despite that he was fully aware of the fact that a competitor had something better to offer. Did I tell the customer to take his business elsewhere after he agreed to our proposition? Hell no, that would have cost me my commission!


Managers stretch their integrity as well, almost daily:

An employee asks about his career chances. The manager answers:”In our organization, everything is possible for ambitious and motivated people.” It is a carrot on a stick, instead of telling how it is out of fear that the employee might leave.

Relaying a not so popular message to their employees, emphasizing that they don’t like it either, but it’s the boss’ decision. That’s unethical and cowardly behavior, not to say that it shows very poor management skills.

These are just a few examples of stretching integrity. We start selling our integrity when our ‘normal’ level of integrity will cost us money, respect, reputation or whatever is of value to us.

I don’t trust on somebody’s integrity, mainly because I don’t trust my own integrity. I know for sure if I had to choose between my principles and integrity on one side and going broke on the other side, my principles and integrity would fly out of the window faster than you can blink your eyes. And I bet that the same goes for the vast majority of us.

Don’t Buy an Integration Tool Without This Functionality

Enterprises are buying integration solutions to combine new and old systems and simplify their IT environment. However they are recklessly embracing new technologies without evaluating the integration functionalities. Combining a highly convoluted legacy IT environment where a mix of cloud and on premise applications are running in parallel is not possible without any-to-any integration. Here are some benefits that you get with the tool.

Makes You Easier to Do Business With: Plays a critical role in smoothing electronic data interchange with partners and clients. Leverages a no-code self service integration approach that reduces customer data on-boarding by 70%.

Puts IT in a Governance Role: Provides a central platform where IT assumes a governance role and business users can handle operational load. Allows IT to focus on other important roles. IT teams can focus on building integration and business users can build integrations faster.

Optimizes Resources and Lowers Cost: Delivers faster go to market capabilities that accelerate time to revenue. Drives down operational cost by 80% and increases profit margins by 20%. Less coding means less cost involved and improved time to revenue. Businesses can transact documents faster and lower the cost involved significantly.

Foundational support to IT: Scales to address pervasive integration needs. The enterprise class integration solution handles any-to-any integration scenarios. It performs polyglot programming for combining systems. This means businesses can transact structured, unstructured files with top most integrity.

Faster Data Deployment and Safer Data Exchange: Provides secure, end-to-end encrypted environment for all data that is transferred and exchanged between partners. Pre-built application connectors ensure secure electronic data interchange across hybrid IT environments.

Your Source of Power, Magic and Miracles

Integrity is often looked at as moral, as making good things in the world and doing the “right” things. This view is a result of our making choices in life based on looking at them as “good vs. bad” or “right vs. wrong” or “true vs. false.” Most of the time these choices arise from feelings of guilt, shame, blame or at our darkest moment when we have been accused of not having integrity. This is a very limited way of thinking and approaching our lives. In fact, integrity is a way to empower ourselves and others.

Who gives us the power and permission to judge and evaluate others’ integrity? What gives us the right to base others’ level of integrity on what we think integrity is or should be? We judge and evaluate others’ decisions, their lifestyle or life choices and anything else we can push our opinions on, so we can impose our idea of integrity on them. When they don’t accept our viewpoint, we become upset and resentful. We distance ourselves from them. When considering your idea of integrity, I invite you to separate your view of social law, religious beliefs and/or cultural beliefs from your idea of integrity at this moment. Look at integrity as a personal phenomenon.

Every person declares his or her own personal integrity based on personal beliefs, faith, values, principles and life choices. However, we all have one thing in common when it comes to integrity and that is… without it things do not work well in our lives and in our commitments! Whatever our personal integrity is, when we do not practice it we do not accomplish our goals neither, therefore, our intentions. Integrity is not constrained or limited by rules, agreements, descriptions, in-order-to’s or demands to get anywhere or to make ourselves or someone else do anything.

Integrity is the ability to be as good as our word, to declare power over our promises, to be true to our personal values and principles, to live our lives as our own people. Ultimately, being true to ourselves and being true to our self-expression is the simplest form of integrity. That is the only force in life that will bring us power, joy, freedom and happiness. We will fail if we live our lives and hold up to integrity only for others or to get others’ approval, to please someone else or impress another human being. A life lived with integrity is a life that is based on your personal values and principles.

You can use your integrity as a meter, like a sensor to distinguish and realize your degree of the reliability or truthfulness of your actions to your commitments and, ultimately, your word. To me, integrity is when we consistently produce what we promise to produce, do what we say we will do, or say what we tell others we will say. This rule applies to what we promise ourselves and to others. Regardless of the circumstances or obstacles in your way, personal integrity is one of powerful forces of decision making. Why? Notice that each time we get into trouble or cause upset with ourselves or others, it is when we don’t follow through with our personal values and principles or what we stand for in life. When we sell ourselves and our integrity short for the sake of being liked, being part of the crowd and/or being ordinary, we short ourselves.