Derivatives and Integrals of Inverse Trig Functions

This section covers:

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


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

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

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


Let’s try some problems:

Trig Inverse Differentiation


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:


Inverse Trig Indefinite Integration

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

Inverse Trig Indefinite Integration 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.

Inverse Trig Definite Integration 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!