• home
• sitemap
• LiveMath info
• Flash highlights
• Scientific Notebook
• math blog
• about
• feedback

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?

Follow IntMath on Twitter

Get the Daily Math Tweet!
IntMath on Twitter

Recommendation

Easy to understand calculus lessons on DVD. Try before you commit. More info:
MathTutorDVD.com

From the math blog...

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

---

Similarly, it can be shown that

Hence, we can summarise that :

---

Example 4: Integrate:

Answer

---

Example 5: Integrate:

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.

Answer

---

2.

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:

Here is the LiveMath solution to this problem.

LIVEMath

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