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- Integral Calc – A Look at Calculus Integration
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## Integral Calc – A Look at Calculus Integration

While Calculus I is primarily devoted to differential calculus or the study of derivatives, most of Calculus 2 and later focuses on integral calculus based on the study of integrals and the process of integration. Whole courses are devoted to integration because it is such a critical operation in mathematics, and in calculus there are many different methods and techniques used for integration in different situations. Here we 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. The indefinite integral is simply the antiderivative of a function and is the function itself. The definite integral finds the difference between two specific values of the indefinite integral and usually gives a numerical answer instead of a function. Definite integrals can be used to find areas and volumes of irregular shapes that cannot be found with basic geometry if the sides of the shape being measured follow some function that can be integrated. For example, a definite integral of x² from 0 to 3 would find the area between the x-axis and the parabola from 0 to 3. This shape is like a triangle with a hypotenuse parabola curve, and is a great example of quickly finding the area of an irregular two-dimensional shape using a definite integral.

In calculus, you learn that the chain rule is the basic rule for taking derivatives. Its counterpart in integral calculus is the process of integration by substitution, also known as u-substitution. In general, if you try to take the integral of a function of the form f(g(x)) * g'(x), the result is simply f(x). However, this general theme has several variations and can even be extended to handle multivariate functions. For a basic example, suppose you want to find the indefinite integral of (x+1)² dx. Let u = x+1 and du = dx. After substituting u for x+1 and du for dx, what remains is to try to take the integral of u² du, which based on our basic patterns is simply u³/3 + C. We substitute x+. 1 back for u in our final answer and quickly (x+1)³/3 + C.

Computational integration is often thought of as a strategic process rather than a straightforward mechanical process because you have many tools at your disposal to integrate features. One very important tool is piecemeal integration, which is a product differentiation rule game. In short, if there are two functions, let’s call them u and v, then the integral of u dv is equal to the integral of v du of the function uv. This may seem like just another random formula, but its importance is that it often allows us to simplify the function we are taking as an integral. This strategy requires that we choose u and du in such a way that the derivative of u is less complex than u. When we break the integral into parts, the integral we get contains du but not u, which means that the function we are taking the integral has simplified in the process.

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