Step-by-Step Guide to Exercise 14.3: Integrating Trigonometric Functions of Class 12

Hi everyone! welcome back another video today i am going to reveal notes of Antiderivative Exercise 14.3 let's ready to levels up your knowledge
Let’s break this down into simple terms for class 12 students.
Definition of Standard Integration of Trigonometric Functions
*Integration* is a method in mathematics used to find the area under a curve, the accumulated total, or to recover a function from its derivative. When we deal with *trigonometric functions* like sine (sin), cosine (cos), and tangent (tan), there are specific rules that help us integrate these functions more easily.
Important Points to Remember
1. **Basic Trigonometric Integrals**: There are standard forms for integrating basic trigonometric functions. Here are some of the most common ones:
*\(\int \sin(x) \, dx = -\cos(x) + C\)*
*\(\int \cos(x) \, dx = \sin(x) + C\)*
*\(\int \sec^2(x) \, dx = \tan(x) + C\)*
*\(\int \csc^2(x) \, dx = -\cot(x) + C\)*
*\(\int \sec(x) \tan(x) \, dx = \sec(x) + C\)*
*\(\int \csc(x) \cot(x) \, dx = -\csc(x) + C\)*
Here, \(C\) represents the **constant of integration**, which accounts for all possible vertical shifts of the function.
2. **Understanding the Function's Behavior**: Knowing how trigonometric functions behave helps in integration. For example:
The sine function oscillates between -1 and 1.
The cosine function shifts it by a quarter turn (90 degrees).
Tangent has vertical asymptotes (it goes to positive or negative infinity) where the cosine function is zero.
3. **Using Identities**: Trigonometric identities can simplify complex integrals. For example, using the identity \(\sin^2(x) + \cos^2(x) = 1\) can help to change the function into a more integrable form.
4. **Substitution Method**: Sometimes, we can simplify integrals using the substitution method. If we have a function like \(\int \sin^2(x) \, dx\), we can use the identity to express it in a different form (like using the half-angle identity).
5. **Follow the Steps**:
Identify the trigonometric function you want to integrate.
Check if it matches one of the standard forms.
If needed, apply trigonometric identities or the substitution method to simplify.
Finally, add the constant of integration \(C\).
Example Problem
Let’s look at an example to put these points into practice.
**Problem**: Integrate \(\int \sin(x) \, dx\).
**Step-by-Step Solution**:
1. Recognize that \(\sin(x)\) matches one of our standard forms.
2. Use the standard integral: \(\int \sin(x) \, dx = -\cos(x) + C\).
3. Write the answer: \(-\cos(x) + C\).
Conclusion
Integration of trigonometric functions is a fundamental skill in calculus. By remembering these standard forms and techniques, you'll find it easier to tackle integration problems involving trigonometric functions. Happy learning!
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Step-by-Step Guide to Exercise 14.3: Integrating Trigonometric Functions

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❤next chapters exercise also solution sir 14.4

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