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AP Calculus BC Intrgrals of Trigonometry functions

Integrals of Trigonometric Functions

Learning Objectives

By the end of this lesson, you will be able to:

Recall the basic trigonometric integration formulas.
Evaluate integrals of sine, cosine, tangent, cotangent, secant, and cosecant functions.
Use trigonometric identities to simplify and integrate more complex trigonometric expressions.
Understand how derivatives and integrals of trigonometric functions are related.

Intrgrals of Trigonometry functions are reverse of Trigonometry functions.

Derivatives of Trigonometric Functions

Let’s start with what we already know. the trigonometry functions derivative is :

Function	Derivative
$$\sin(x)$$	$$\cos(x)$$
$\cos(x)$	$$-\sin(x)$$
$$\tan(x)$$	$$\sec^{2}(x)$$
$$\cot(x)$$	$$-\csc^{2}(x)$$
$$$\sec(x)$$$	$$\sec(x)\tan(x)$$
$$$\csc(x)$$$	$$-\csc(x)\cot(x)$$

The integrals will be the reverse of these.

Basic Integrals of Trigonometric Functions

Function	$Integral$
$$\sin(x)$$	$$-\cos(x)+C$$
$$\cos(x)$$	$$\sin(x)+C$$
$$\sec^{2}(x)$$	$\tan(x)+C$
$$\csc^{2}(x)$$	$-\cot(x)+C$
$$\sec(x)\tan(x)$$	$$\sec(x)+C$$
$$$\csc(x)\cot(x)$$$	$-\csc(x)+C$

And Trigonometry functions has coefficient integrals is

Function	$Integral$
$$\sin(ax)$$	$$-\frac{cos(ax)}{a}+C$$
$$\cos(ax)$$	$$\frac{sin(ax)}{a}+C$$
$$\sec^{2}(ax)$$	$\frac{tan(ax)}{a}+C$
$$\csc^{2}(ax)$$	$-\frac{cot(ax)}{a}+C$
$$\sec(ax)\tan(ax)$$	$$\frac{sec(ax)}{a}+C$$
$$$\csc(ax)\cot(ax)$$$	$-\frac{csc(ax)}{a}+C$