21. Antiderivatives, Areas and the FTC

b. Antiderivative Rules

3. Trig Functions

The table on the previous page gave the antiderivatives for exponential and logarithmic functions. The tables on this page give the antiderivatives for the trigonometric functions and their inverses.

Derivatives

We first review their derivatives:

Trig Functions	Inverse Trig Functions
\(\dfrac{d}{dx}\sin x=\cos x\)	\(\dfrac{d}{dx}\arcsin x=\dfrac{1}{\sqrt{1-x^2}}\)
\(\dfrac{d}{dx}\cos x=-\sin x\)	\(\dfrac{d}{dx}\arccos x=-\,\dfrac{1}{\sqrt{1-x^2}}\)
\(\dfrac{d}{dx}\tan x=\sec^2 x\)	\(\dfrac{d}{dx}\arctan x=\dfrac{1}{1+x^2}\)
\(\dfrac{d}{dx}\cot x=-\csc^2 x\)	\(\dfrac{d}{dx}\,\mathrm{arccot}\,x=-\,\dfrac{1}{1+x^2}\)
\(\dfrac{d}{dx}\sec x=\sec x\tan x\)	\(\dfrac{d}{dx}\,\mathrm{arcsec}\,x=\dfrac{1}{|x|\sqrt{x^2-1}}\)
\(\dfrac{d}{dx}\csc x=-\csc x\cot x\)	\(\dfrac{d}{dx}\,\mathrm{arccsc}\,x=-\,\dfrac{1}{|x|\sqrt{x^2-1}}\)

Antiderivatives

Here are their antiderivatives. You can check each line of the table by differentiating the quantity on the right to see you get the quantity on the left.

Trigonometric Functions

	If the Function is	then the Antiderivative is
Sine	\(f(x)=\cos x\)	\(F(x)=\sin x+C\)
Cosine	\(f(x)=\sin x\)	\(F(x)=-\cos x+C\)
Tangent\(^\text{1}\)	\(f(x)=\sec^2 x\)	\(F(x)=\tan x+C\)
Cotangent\(^\text{1}\)	\(f(x)=\csc^2 x\)	\(F(x)=-\cot x+C\)
Secant\(^\text{1}\)	\(f(x)=\sec x\tan x\)	\(F(x)=\sec x+C\)
Cosecant\(^\text{1}\)	\(f(x)=\csc x\cot x\)	\(F(x)=-\csc x+C\)

\(^\text{1}\) These are the functions whose antiderivatives are the trig functions: \(\tan x\), \(\cot x\), \(\sec x\) and \(\csc x\). We will discuss the antiderivatives of these trig functions in Calculus 2 in the chapter on Trig Integrals.

Inverse Trigonometric Functions

	If the Function is	then the Antiderivative is
Inverse Sine\(^\text{2}\)	\(f(x)=\dfrac{1}{\sqrt{1-x^2}}\)	\(F(x)=\arcsin x+C\)
Inverse Tangent\(^\text{2}\)	\(f(x)=\dfrac{1}{1+x^2}\)	\(F(x)=\arctan x+C\)
Inverse Secant\(^\text{2}\)	\(f(x)=\dfrac{1}{x\sqrt{x^2-1}}\)	\(F(x)=\mathrm{arcsec}\,x+C\)
Inverse Cosine\(^\text{3}\)	\(f(x)=\dfrac{1}{\sqrt{1-x^2}}\)	\(F(x)=-\arccos x+C\)
Inverse Cotangent\(^\text{3}\)	\(f(x)=\dfrac{1}{1+x^2}\)	\(F(x)=-\,\mathrm{arccot}\,x+C\)
Inverse Cosecant\(^\text{3}\)	\(f(x)=\dfrac{1}{x\sqrt{x^2-1}}\)	\(F(x)=-\,\mathrm{arccsc}\,x+C\)

\(^\text{2}\) These are the functions whose antiderivatives are the inverse trig functions: \(\arcsin x\), \(\arctan x\) and \(\mathrm{arcsec}\,x\). We will discuss the antiderivatives of these inverse trig functions in Calculus 2 in the chapter on Integration by Parts.

\(^\text{3}\) These antiderivative formulas are not useful, since the formulas using \(\arcsin x\), \(\arctan x\) and \(\mathrm{arcsec}\,x\) are easier. They just differ by a constant.