Integrals of Trigonometric Functions
Learning Objectives
By the end of this lesson, you will be able to:
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Recall the basic trigonometric integration formulas.
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Evaluate integrals of sine, cosine, tangent, cotangent, secant, and cosecant functions.
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Use trigonometric identities to simplify and integrate more complex trigonometric expressions.
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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 |
|---|---|
| $$\sin(x)$$ | $$\cos(x)$$ |
| $$\cos(x)$$ | $$-\sin(x)$$ |
| $$\tan(x)$$ | $$\sec^{2}(x)$$ |
| $$\cot(x)$$ | $$-\csc^{2}(x)$$ |
| $$\sec(x)\tan(x)$$ | |
| $$-\csc(x)\cot(x)$$ |
The integrals will be the reverse of these.
Basic Integrals of Trigonometric Functions
| Function | |
|---|---|
| $$\sin(x)$$ | $$-\cos(x)+C$$ |
| $$\cos(x)$$ | $$\sin(x)+C$$ |
| $$\sec^{2}(x)$$ | $$\tan(x)+C$$ |
| $$\csc^{2}(x)$$ | $$-\cot(x)+C$$ |
| $$\sec(x)\tan(x)$$ | $$\sec(x)+C$$ |
| $$-\csc(x)+C$$ |
And Trigonometry functions has coefficient integrals is
| Function | |
|---|---|
| $$\sin(ax)$$ | $$-\frac{cos(ax)}{a}+C$$ |
| $$\cos(ax)$$ | $$\frac{sin(ax)}{a}+C$$ |
| $$\sec^{2}(ax)$$ | $$\frac{tan(ax)}{a}+C$$ |
| $$\csc^{2}(ax)$$ | $$-\frac{cot(ax)}{a}+C$$ |
| $$\sec(ax)\tan(ax)$$ | $$\frac{sec(ax)}{a}+C$$ |
| $$-\frac{csc(ax)}{a}+C$$ |