Derivatives and Integrals of Inverse Trig Functions

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

Let’s try some problems:

Trig Inverse Differentiation

Integrals of the Inverse Trig Functions

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

Here are the integration formulas for 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 Of Inverse Trig Functions

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.

Let’s first do some Inverse Trig Indefinite Integration problems:

Inverse Trig Indefinite Integration