This section covers:

**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**

**Antiderivatives** are just the opposite of derivatives; it’s that simple. For example, since we know that the derivative of 5*x* is 5, the antiderivative of 5 is 5*x*. 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 **5 x** is “

**5 + C**”, where

**C**is any constant.

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!

# Indefinite Integration Problems

Let’s do 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.

# 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** of each other. But sometimes we know more about the curve, so we can determine a **particular solution** (get what the C is), and we can do this say with one (*x*, *y*) value on the curve, which is called the **initial condition**.

So basically what you do is 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**.

Let’s do 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 here in the **Definite Integration** 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, it’s
**position**(relative to say the origin) is a function, but the**derivative**of this function would be its (**instantaneous**)**velocity**(how fast it’s moving) at a certain point, and the**derivative of its velocity**would be 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 a**displacement**, 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. - The
**velocity**function is the**derivative**of the position function, and 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).

Let’s do some problems:

Here’s one more problem:

**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! **