This section covers:

**Derivatives of the Inverse Trig Functions****Integrals Involving the Inverse Trig Functions****More Practice**

We learned about the **Inverse Trig Functions** here, and it turns out that the derivatives of them are not trig 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:

**Derivatives of Inverse Trig Functions**

\(\displaystyle \frac{{d\left( {\arcsin u} \right)}}{{dx}}=\frac{{{u}’}}{{\sqrt{{1-{{u}^{2}}}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{d\left( {\arccos u} \right)}}{{dx}}=\frac{{-{u}’}}{{\sqrt{{1-{{u}^{2}}}}}}\)

\(\displaystyle \frac{{d\left( {\arctan u} \right)}}{{dx}}=\frac{{{u}’}}{{1+{{u}^{2}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{d\left( {\text{arccot }u} \right)}}{{dx}}=\frac{{-{u}’}}{{1+{{u}^{2}}}}\)

\(\displaystyle \frac{{d\left( {\text{arcsec }u} \right)}}{{dx}}=\frac{{{u}’}}{{\left| u \right|\sqrt{{{{u}^{2}}-1}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{d\left( {\text{arccsc }u} \right)}}{{dx}}=\frac{{-{u}’}}{{\left| u \right|\sqrt{{{{u}^{2}}-1}}}}\)

# Integrals Involving the Inverse Trig 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 involving the Inverse Trig Functions; notice that we only have formulas for **three of the inverse trig functions**; trust me, it works this way!

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-substitution to solve the problems.

**Integrals Involving the Inverse Trig Functions**

\(\displaystyle \begin{align}\int{{\frac{{du}}{{\sqrt{{1-{{u}^{2}}}}}}}}\,=\arcsin \,u+C\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int{{\frac{{du}}{{\sqrt{{{{a}^{2}}-{{u}^{2}}}}}}}}\,=\arcsin \,\frac{u}{a}+C\\\int{{\frac{{du}}{{1+{{u}^{2}}}}}}\,=\arctan \,u+C\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int{{\frac{{du}}{{{{a}^{2}}+{{u}^{2}}}}}}\,=\frac{1}{a}\arctan \,\frac{u}{a}+C\\\int{{\frac{{du}}{{u\sqrt{{{{u}^{2}}-1}}}}}}\,=\text{arcsec}\,\left| u \right|+C\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int{{\frac{{du}}{{u\sqrt{{{{u}^{2}}-{{a}^{2}}}}}}}}\,=\frac{1}{a}\text{arcsec}\,\frac{{\left| u \right|}}{a}+C\end{align}\)

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, just 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.

Let’s first do some **Indefinite Integration** problems:

Here are a few more integrals involving inverse trig functions that are bit more complicated:

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.

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