Prerequisites

To keep this article as accessible as possible, the prerequisites are minimal. A basic understanding of algebra (high school level), simple trigonometry, and ideally, some knowledge of differential calculus will suffice. For those needing a refresher on differential calculus, a separate article is available that is concise and informative.

Formatting & Structure

This article will encompass various topics that collectively enhance your integration skills. You should be equipped to tackle most standard integrals, including those commonly found in competitive math contexts. A future article will address competition-level integrals, which typically involve familiar techniques. Additionally, we strive to provide intuitive understanding through examples and brief exercises for practice.

It's also important to note that in this text, "log x" refers to the natural logarithm. The content is organized as follows:

• Riemann integration & its intuition
• Fundamental Theorem of Calculus
• Basic properties of integration
• Substitution techniques for integrals & valuable tricks
• Integration by parts
• Partial fraction decomposition
• Advanced substitution methods in integration

Now, let’s dive right in!

Riemann Integration

Imagine we want to calculate the area under a graph, for instance, the line represented by y = x, between x = 0 and x = 2. The area beneath this graph forms a triangle, which can be easily calculated as half the area of a rectangle measuring 2 by 2. Hence, the area is simply 2.

What happens, however, if we wish to determine the area under a more complex function, such as y = 3x²? The graph appears more intricate, and standard geometric shapes will not suffice for calculation. Instead, we can estimate the area using a series of equally spaced rectangles.

Let’s initially use two rectangles. The area estimate would then be approximately:

0.75 * 0.5 + 0.5 * 3 = 1.875.

However, this estimate is quite rough! We can improve it by increasing the number of rectangles. If we opt for four rectangles, the estimation becomes more accurate. As we continue to increase the number of rectangles and reduce their width, our estimate will converge to the actual area beneath the curve.

Formally, we express this by defining a partition of the interval [a, b], where we let a = x₀ and b = xₙ. As the number of rectangles approaches infinity, we can sum their areas to find the total area under the curve.

If the distances between the partitions aren’t equal, the method remains valid, allowing for varied approaches to calculating limits. Let’s return to our example of y = 3x², applying the partitioning method to find the area under the graph.

The area estimation leads us to a sum that can be simplified using known formulas. After some algebra, it can be shown that the area beneath the curve from 0 to 1 equals 1.

The Fundamental Theorem of Calculus

To streamline our calculations, we adopt the concept of a definite integral, which represents the signed area of f(x) between limits a and b. This area can be positive or negative, depending on whether the function is above or below the x-axis.

The relationship between integration and differentiation is established by the Fundamental Theorem of Calculus, which states that integration is essentially the reverse process of differentiation. This leads us to the concept of an indefinite integral, where we remove limits and introduce a constant of integration.

Basic Integration Properties

Having established that integration is the inverse of differentiation, we can conclude that constants can be factored out of integrals, and we can sum integrals over different intervals.

Moving forward, we will explore basic integrals and recall differentiation properties. For instance, the integral of 1/x leads us to the natural logarithm, log x, which serves as a crucial anti-derivative.

Integration by Substitution

In our differentiation article, we examined the chain rule, prompting the question: can we reverse this process for integration? Indeed, we can! This is commonly achieved through a u-substitution.

If we define u = f(x) and differentiate, we can express dx in terms of du, allowing us to integrate in terms of u. It's essential to convert back to x when concluding our integration unless we are working with limits.

Trigonometric substitutions are useful in specific scenarios, particularly when we encounter recognizable identities.

Integration by Parts

While we cannot multiply integrals directly, we can draw parallels from differentiation. By applying integration by parts, we can simplify integrals by rearranging the functions involved.

Through strategic selection of u and v, we can reduce complicated integrals to more manageable forms.

Method of Partial Fractions

When faced with integrals that resist traditional substitution or parts, we can employ partial fraction decomposition. By breaking down complex fractions into simpler components, we can integrate each part effectively.

Advanced Substitution Techniques

Finally, we touch on advanced substitutions, including the Weierstrass substitution, which transforms trigonometric integrals into rational expressions.

This concludes our introductory exploration of integration. There are many more techniques and integrals to discover, and I encourage you to seek out additional resources to further your understanding, such as the MIT or UK integration bees.

Thank you for engaging with this material! Your support is appreciated, and I welcome suggestions for future topics.

In this first video, "An Introduction to Integration," you'll gain foundational insights into integral calculus concepts.

The second video, "Basic Integration... How? (NancyPi)," provides practical examples and techniques for solving integrals.