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4. Integration: Basic Trigonometric Forms

by M. Bourne

In this section, we obtain the following integral formulas by reversing the formulas for differentiation of trigonometric functions that we met earlier:

We now apply the power formula to integrate some examples.

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Example 1: Integrate:

Answer

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Example 2: Integrate:

Answer

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Integral formulas for some other trigonometric functions: (These are obtained by just reversing the differentiation process).

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Example 3: Integrate:

Answer

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NOTE: The above formulas were obtained by reversing the differentiation process.

 

Further Integrals

Now, if we want to find , we note that

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Similarly, it can be shown that

Hence, we can summarise that :

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Example 4: Integrate:

Answer

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Example 5: Integrate:

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Example 6: Find the area under the curve of y = sin x from x = 0 to .

Answer

 

Exercises

Integrate each of the given functions:

1.

Answer

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2.

Answer

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3. If the current in a certain electric circuit is , find the expression for the voltage across a 500-mF capacitor as a function of time. The initial voltage is zero. Show that the voltage across the capacitor is 90°out of phase with the current.

Here is the LiveMath solution to this problem.

LIVEMath

Answer

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4. A force is given as a function of the distance from the origin as

Express the work done by this force as a function of x if W = 0 for x = 0.

Answer

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Chapter Contents

• Methods of Integration
• 1. Integration: The General Power Formula
• 2. Integration: The Basic Logarithmic Form
• 3. Integration: The Exponential Form
• 4. Integration: The Basic Trigonometric Forms
• 5. Integration: Other Trigonometric Forms
• 6. Integration: Inverse Trigonometric Forms
• 7. Integration by Parts
• 8. Integration by Trigonometric Substitution
• 9. Integration by Use of Tables
• Table of Common Integrals
• 10. Integration by Reduction Formulae
• 11. Integration by Partial Fractions

• Comments, Questions?

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