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## Integral Calc – A Look at Calculus Integration

While Calculus I is primarily dedicated to differential calculus, or the study of derivatives, most calculations 2 and beyond focus on integral calculus, which is based on the study of integrals and the process of integration . Integration has entire courses dedicated to it because it is such a critical operation in mathematics, and there are many different integral calculus methods and techniques that are used for integration in different situations. Here we will look at an overview of some of these techniques and the types of integrals that can be taken.

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 usually 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, as long as the sides of the measured figure follow a 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 of 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 you try 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 set u = x+1, and du = dx. After replacing u in place of x+1 and du in place of dx, we only have to try to take the integral of u² from du, which we know from our basic models to be is just u³/3 + C. We substitute x+ 1 back for you in our final answer, and quickly get (x+1)³/3 + C.

Computing integration is often seen as a strategic process instead of a simple mechanical process due to the large number of tools available to you for integrating functions. A very important tool is integration by parts, which plays on the product rule for differentiation. In short, if there are two functions, call them u and v, then the integral of u dv is equal to uv – the integral of v du. It may seem like just another random formula, but its significance is that it often allows us to simplify a function whose integral we are taking. This strategy requires that we choose u and du so that the derivative of u is less complicated than u. Once we break the integral down into parts, our resulting integral contains du, but not u, which means that the function whose integral we are taking has simplified in the process.

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