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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.

Example 1: Integrate:

Answer

Example 2: Integrate:

Answer

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

Example 3: Integrate:

Answer

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

Further Integrals

Now, if we want to find , we note that

math expression

Similarly, it can be shown that

Hence, we can summarise that :

Example 4: Integrate:

Answer

Example 5: Integrate: math expression

Answer

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. math expression

Answer

2. math expression

Answer

3. If the current in a certain electric circuit is i = 110 cos 377t, find the expression for the voltage across a 500-μF 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.

We need the following result from electronics, which gives the voltage across a capacitor, where C is the capacitance:

Answer

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

3. Integration: The Exponential Form

5. Integration: Other Trigonometric Forms

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