# Integration By Trigonometric Substitution

TLDRThe script discusses integration by trigonometric substitution, a technique for integrals containing one of three specific radical expressions. It explains how making trigonometric substitutions for x simplifies these expressions via Pythagorean identities. An example shows the process: make the substitution, convert dx into dθ, integrate with respect to θ, then convert θ back to x using a labeled triangle. Two more examples demonstrate applying the technique. The video teaches three key integration tricks - substitution rule, integration by parts, and now trigonometric substitution - and notes that determining which to apply takes practice.

###### Takeaways

- 📚 Trigonometric substitution is a technique for integrating expressions containing √(A² - x²), √(A² + x²), or √(x² - A²), where A is any positive number.
- 📈 It differs from the substitution rule and integration by parts, filling a gap when these methods are not applicable due to the absence of a suitable du term.
- 🤖 Specific substitutions are required: x = A sin(θ) for √(A² - x²), x = A tan(θ) for √(A² + x²), and x = A sec(θ) for √(x² - A²).
- ✅ These substitutions simplify the integrand by leveraging Pythagorean identities, transforming the radical expressions into simpler trigonometric functions.
- ⏱️ Converting dx to dθ is crucial, requiring differentiation of the substitution expression to proceed with integration in terms of θ.
- 🔬 The technique involves three key steps: identifying the specific radical expression, performing the trigonometric substitution and simplification, and integrating the resulting expression.
- 🌐 After integration, results in terms of θ are converted back to x using trigonometric identities and the geometry of a right-angled triangle.
- ✔️ This method is particularly useful for integrands that are not directly integrable using standard techniques, enabling integration of complex radical expressions.
- 📝 Practice and familiarity with trigonometric identities and substitutions are essential for effectively applying trigonometric substitution.
- 👀 The choice between substitution, integration by parts, and trigonometric substitution depends on the form of the integrand, highlighting the importance of strategy in integration.

###### Q & A

### What are the three specific terms that allow trigonometric substitution to be used?

-The three specific terms are: the square root of A squared minus x squared, the square root of A squared plus x squared, and the square root of x squared minus A squared.

### What is the substitution that must be made when the term 'square root of A squared minus x squared' is present?

-When the term 'square root of A squared minus x squared' is present, x must be substituted with A sine theta.

### Why must dx be changed to dtheta after making the trigonometric substitution?

-Dx must be changed to dtheta because the substitution changes everything to be in terms of theta. To integrate with respect to theta, there must be a dtheta term rather than dx.

### What are the three main steps when using trigonometric substitution?

-The three main steps are: 1) Recognize which of the three expressions allows substitution and make the appropriate substitution. 2) Manipulate dx to get an expression with dtheta. 3) Construct a triangle to change the integral back into terms of x after integrating with respect to theta.

### When can the substitution rule be used vs when can trig substitution be used?

-The substitution rule can be used when the integrand has a term that can act as du. Trig substitution is used when one of the three specific radical terms is present and the substitution rule does not apply.

### What does the procedure look like to change the integral solution back to terms of x?

-A triangle is constructed using the original substitution. The triangle is labeled so that terms with theta can be converted back into terms with x using trigonometric ratios.

### What is cotangent squared theta equal to in terms of more basic trig functions?

-Cotangent squared theta is equal to cosecant squared theta minus one.

### Why convert cotangent theta to an expression involving x in the first example?

-The original integral was in terms of x, so the final solution needs to be as well. Converting cotangent theta allows the solution to be written with the original variable x rather than theta.

### What additional method had to be used to evaluate the integral in the second example?

-After the trig substitution, another substitution of u=sine theta was required. This allowed the integral to be evaluated using basic substitution methods.

### Why are three substitution techniques useful to know?

-Having multiple integration techniques available allows for determining the best approach based on the form of the integrand. Each technique handles certain forms particularly well.

###### Outlines

##### 📝 Introducing Trigonometric Substitution

Paragraph 1 introduces trigonometric substitution, a technique for integrating certain functions containing radicals. It explains that trigonometric substitution works when the integrand contains one of three specific terms - the square root of A^2 - x^2, A^2 + x^2, or x^2 - A^2. For each of these three cases, a specific trigonometric substitution must be made - x = A sinθ, x = A tanθ, or x = A secθ.

##### 😊 Walkthrough of Trig Substitution Example

Paragraph 2 walks through a specific example of using trigonometric substitution to integrate √(9 - x^2) / x^2 dx. It substitutes x = 3sinθ, converts dx to dθ, simplifies the integrand, performs the integration, and converts the final result back in terms of x.

##### 👍 Another Trig Substitution Example

Paragraph 3 provides another example of applying trigonometric substitution to integrate 1 / [x^2 * √(x^2 + 4)] dx. It substitutes x = 2tanθ, simplifies, integrates, and converts back to x. This illustrates the full process again.

###### Mindmap

###### Keywords

##### 💡integration

##### 💡substitution rule

##### 💡integration by parts

##### 💡trigonometric substitution

##### 💡Pythagorean identities

##### 💡hypotenuse

##### 💡integrand

##### 💡adjacent

##### 💡opposite

##### 💡comprehension

###### Highlights

First significant research finding

Introduction of innovative methodology

Key conclusion with practical applications

###### Transcripts

5.0 / 5 (0 votes)

Thanks for rating: