Antiderivatives | Initial Conditions and Particular Solutions |

Basic Integration Rules | Position, Velocity, and Acceleration |

Trigonometric Integration Rules | More Practice |

Indefinite Integration Problems |

**WARNING**: The techniques in this section only work if the argument of what’s being integrated is just “$ x$”; in other words, “$ x$” is by itself and doesn’t have a coefficient or perhaps more complicated. See the **U-Substitution Integration** section for more integrating more complicated expressions!

**Antiderivatives (Anti-derivatives)**

**Antiderivatives** are just the opposite of derivatives; it’s that simple. For example, since we know that the derivative of $ 5x$ is **5**, the antiderivative of **5** is $ 5x$. But we have to be careful here, since the derivative of $ 5x+4$ is also **5** (since the derivative of a constant is **0**). So, technically, the antiderivative of **0** is “$ C$”, where $ C$ is any constant, because the derivative of $ C$ is **0**.

When we take the antiderivative of something, we are actually integrating it, or taking the **integral **of it. The **integrand** is the name of what we’re taking the integral of.

Here are definitions of an **antiderivative**, **integral**, **constant of integration**, and **differential equation. **We’ll talk more about **Differential Equations** in the **Differential Equations and Slope Fields** section.

## Basic Integration Rules

I know this seems confusing at this point, but let’s go through some basic integration rules and examples, and then do some problems!

## Trigonometric Integration Rules

Here are the **trigonometry** integration rules:

Here are some **hints** to help you remember the trig differentiation and integration rules: When the trig functions start with “**c**”, the differentiation or integration is **negative** (**cos** and **csc**). For the functions other than **sin** and **cos**, there’s always either **one tan** and **two secants**, or **one cot** and **two cosecants** on either side of the formula. Look at the formulas and see how this makes sense!

**Note**: More trig integration rules (involving “**ln**”) will be introduced later here in the **Exponential and Logarithmic Integration **section. These are the following:

## Indefinite Integration Problems

Here are some problems; notice how we may need to **rewrite** the integral, and then **simplify** at the end. Note to check your work, you can **differentiate back from the answer** to see if you get the original! Also, remember that you can check these by going to **www.wolframalpha.com** (or use app on smartphone) and type in “**integral of x^.25**” for example.

Here are more problems:

## Initial Conditions and Particular Solutions

Since when we take an integral of a function, we have to add the “$ +\,C$” part to it (the **constant of integration**), if we were to graph the solutions, we’d have many different curves that are the same but **Vertical Translations **(here in the** Parent Functions and Transformations **section) of each other. Sometimes we know more about the curve, so we can determine a **particular solution** (get what the $ C$ is), and we can do this, for example, with one $ (x,y)$ value on the curve, which is called the **initial condition**. Thus, we can take the integral with the “$ +\,C$” part, and then put in the **initial condition** values given for $ x$ and $ y$ (which is $ f(x)$), and solve for $ C$. This will give you the **particular solution**.

Note that we have to go through the exercise more than once when we are given the **second derivative**. Here are some examples:

And here’s a **word problem**. Note how in this problem, we have to use a **System of Equations** to solve for the particular solution. Also remember when something is **proportional** to something else, it’s a **Direct Variation**, and one side is the product of a constant $ k$ and the other side.

## Revisiting Position, Velocity, and Acceleration

We talked a little about **Rates of Change and Velocity** here in the **Equation of the Tangent Line, Tangent Line Approximation, and Rates of Change** section (and we will in the **Integration as Accumulated Change** section). Now we’ll be able to work backwards (for example, from a rate to a distance) since we know how to integrate functions! Here are some things we know about position, velocity, and acceleration:

- If a particle is moving along a horizontal line, its
**position**(relative to say the origin) is a function, the**derivative**of this function is its (**instantaneous**)**velocity**(how fast it’s moving) at a certain point, and the**derivative of its velocity**is its**acceleration**(how fast its velocity is changing). - We can take the
**integral**of velocity to get position, and the integral of acceleration to get velocity. - The
**position**of an object is actually a**vector**, since it has both a magnitude (a scalar, such as distance) and a direction. A change in position is the**displacement vector**, which is how far out of place the object is, compared to where it started. The**distance**it has traveled is the total amount of ground an object has covered during its motion; this is the**absolute value of the displacement**, and is a scalar. - The
**velocity**function is the**derivative**of the position function, and can be negative, zero, or positive. If the**derivative**(velocity) is**positive**, the object is**moving to the right**(or up, if that’s how the coordinate system is defined); if**negative**, it’s**moving to the left**(or down); if the**velocity is 0**, the**object is at rest**. This is also called the**direction**of the object. - The
**velocity**of an object is actually a**vector**, whereas the**speed**is the**absolute value of the velocity**, and is a scalar. The**speed**of an object cannot be negative, whereas velocity can. **Acceleration**(the**derivative of velocity**, which is also a vector) can cause speed to increase, decrease, or stay the same. Negative acceleration means slowing down (velocity decreasing) and positive acceleration means speeding up (velocity increasing).

Here are some problems:

Here’s one more problem:

Here are a few more, taken from **Calculus**, 10^{th} Edition, by Ron Larson and Bruce Edwards:

**Understand these problems, and practice, practice, practice!**

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the **Mathway** site, where you can register for the **full version** (steps included) of the software. You can even get math worksheets.

You can also go to the **Mathway** site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!

**On to U-Substitution Integration – you’re ready! **