Here you find exercises and solutions for one-dimensional integration. We explain the rule of substitution, the integration by parts, and more!
Part 1 - Hyperbolic Sine and Cosine
Let’s look at the $\sinh$ and the $\cosh$ functions and let’s calculate their antiderivates by integrating exponentional functions.
Part 2 - Area under a Parabola
A typical exercise in Calculus is to ask about an area between graph and x-axis. Most of the time, this just means that one has to calculate intersection to determine the domain of integration. Let’s look at a simple example of a quadratic function.
Part 3 - Area Between Two Curves
The next step in area calculations is to consider two functions and their graphs. If there are intersection points, then there is a well-defined area between these to graphs, which can be calculated by integration. We will discuss an example with the cosine and sine function.
Part 4 - The Fundamental Theorem of Calculus
The first part of the fundamental theorem of calculus tells us that integrating a continuous function $f$ from a start point to variable end point $x$ always gives us an antiderivative of the function $f$. This fact can be used for calculating derivatives as well.
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Part 5 - Specify a Primitive
The typical application of integration is to determine primitives, also called antiderivatives. Let’s consider an exponential function here.
Part 6 - Exercise 6a (Solving Integrals)
In the next videos, we will apply the toolset about integration we have already learnt. For example, a combination of exponentional and cosine function can be solved by using integration by parts.
Part 7 - Exercise 6b (Solving Integrals)
The next integral can also be tackled by integration by parts.
Part 8 - Exercise 6c (Solving Integrals)
Sometimes there are different possibilities to solve a given integral. We will demonstrate two for the following integral.
Part 9 - Exercise 6d (Solving Integrals)
For the last integral, we will use integration by parts again.
Part 10 - Partial Fraction Decomposition
Another important tool to solve integrals is the so-called partial fraction decomposition, which can be applied to rational functions. Note that a rational function is just a fraction that uses two polynomials. Therefore, is usually needed to calculate the zeros of the denomiator to start the partial fraction decomposition. After it, we can integrate the result term by term.
Part 11 - Substitution
In the next video, we will look at a lot of exercises where we can use the change-of-variable formula, also known as integration by substition.
Part 12 - Improper Integrals
Sometimes we want to calculate an area described by a graph that stretches to infinity. These areas are not represented by ordinary Riemann integrals, but by a limit process. This means we apply our toolset of integration as always, but in the end we have to perform a limit. Of course, the resulting limit can still be a finite number. We speak of improper Riemann integrals in this case. Let’s look at some simple example to demonstrate the workflow.
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Part 13 - Volume of Solid of Revolution
We can see the integration as a procedure to determine an area in the $x-y$-plane. However, when we rotate this area around the $x$-axis, we get a solid body that has a three-dimensional volume. In order to calculate this one, we can also use the integration. Let’s apply it to some exponential function.
Summary of the course Exercises - Integration
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