Derivatives of Inverse Trig Functions | |

Integrals Involving Inverse Trig Functions | |

More Practice |

Note that more generic **Trigonometric Substitution** methods, which use **Right Triangle Trigonometry**, can be found **here**.

We learned about the **Inverse Trigonometric Functions** here, and it turns out that the derivatives of them are not trigonometric expressions, but **algebraic**. When memorizing these, remember that the functions starting with “$ c$” are negative, and the functions with **tan** and **cot** don’t have a square root.

Also remember that sometimes you see the inverse trig function written as $ \arcsin x$ and sometimes you see $ {{\sin }^{{-1}}}x$.

## Derivatives of Inverse Trig Functions

Here are the derivatives of Inverse Trigonometric Functions:

Here are some problems:

## Integrals Involving Inverse Trig Functions (Integrals Resulting in Inverse Trigonometric Functions)

When we integrate **to get Inverse Trigonometric Functions back**, we have use tricks to get the functions to look like one of the inverse trig forms and then usually use **U-Substitution Integration** to perform the integral.

Here are the integration formulas; notice that we only have formulas for **three of the inverse trig functions**; trust me, it works this way (although there may be other correct answers)! To the right of each formula, I’ve included a short-cut formula that you may want to learn; however, if you just know the first formulas at the left (that resemble the differentiation formulas), you will be able to use **U-sub** to solve the problems.

A lot of times, to get the integral in the correct form, we have to play with the function to get a “$ 1$” in the denominator, either in the square root, or without it (for **tan** and **cot**). To do this, take the greatest common factor (**GCF**) of the constant out, so a “$ 1$” will remain; we’ll see this in problems below. Sometimes, we’ll also have to **Complete the Square**, as shown below.

**Note that we don’t really know what the “**$ \boldsymbol {u}$**” is until we factor out the GCF – until after we’ve “made it fit” into one of the formulas above!**

Here are some of the more straight-forward **Indefinite Integration** problems; these can be tricky!

Here are a few more integrals involving inverse trig functions that are bit more complicated. Note that we need to **Complete the Square** in the last problem.

Now let’s do some **Inverse Trig Definite Integration** Problems. Notice in the second problem, we have to use “$ \arcsin x$” for “$ u$”, and we need to **Complete the Square** for the last problem.

**Learn these rules, 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.

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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** Applications of Integration: ****Area and Volume** – you are ready!