Integrals: Trig Substitution 2

Khan Academy
14 Oct 200908:10
EducationalLearning
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TLDRThe video script presents a method for solving a complex indefinite integral using trigonometric identities. The integral in question is the indefinite integral of 1 over (36 + x^2) dx. The process involves rewriting the integral to resemble a standard trigonometric identity, which leads to the substitution of x^2/36 with (tangent of theta)^2. By identifying the derivative of the tangent function and applying the arctangent to simplify the expression, the integral is reduced to a simpler form involving secant squared of theta. Ultimately, the solution is expressed as 1/6 times the arctangent of x over 6, plus a constant (c), demonstrating a clever application of trigonometric principles to resolve a challenging integral.

Takeaways
  • 📚 The problem involves finding the indefinite integral of a complex function without an obvious u-substitution.
  • 🤔 The challenge is to solve the integral of 1 over (36 + x^2) without a straightforward substitution method.
  • 👉 The solution strategy involves using trigonometric identities to simplify the integral.
  • 📈 The integral is rewritten to resemble a form that can be matched with a known trigonometric identity.
  • 🌐 A key trigonometric identity used is 1 + tan^2(θ) = sec^2(θ), derived from the definition of tangent and the unit circle.
  • 🔄 A substitution is made where x^2/36 is equated to tan^2(θ), leading to x = 6tan(θ).
  • 🧠 The derivative of the tangent function is found to be sec^2(θ), using the quotient rule and basic trigonometric relationships.
  • 📝 The integral simplifies to 1/6 dθ after the substitution and cancellation of terms.
  • 🔙 Back substitution is performed using the arctangent function to express θ in terms of x.
  • 🎯 The final answer for the indefinite integral is 1/6 (arctangent(x/6)) + C, where C is the constant of integration.
  • 📊 This approach demonstrates the power of trigonometric identities in simplifying complex integrals and finding antiderivatives.
Q & A

  • What is the integral given in the transcript?

    -The integral given in the transcript is the indefinite integral of 1 over (36 + x^2) with respect to x.

  • Why is the integral difficult to solve without trigonometry?

    -The integral is difficult to solve without trigonometry because there is no straightforward u-substitution that can be applied directly to the given function.

  • What trigonometric identity does the speaker use to simplify the integral?

    -The speaker uses the identity 1 + tan^2(θ) which is equal to sec^2(θ) to simplify the integral.

  • How does the speaker attempt to rewrite the integral to fit a trigonometric identity?

    -The speaker rewrites the integral by placing dx in the numerator and attempts to express it in a form that resembles 1 + something squared, which leads to the use of the trigonometric identity.

  • What substitution is made to simplify the integral?

    -The substitution made is x^2/36 = tan^2(θ), which allows the integral to be expressed in terms of trigonometric functions.

  • What is the derivative of the tangent function with respect to θ?

    -The derivative of the tangent function with respect to θ is sec^2(θ).

  • How does the speaker find the derivative of the tangent function?

    -The speaker finds the derivative of the tangent function by applying the quotient rule to (sin(θ)/cos(θ)) and simplifying the result to get sec^2(θ).

  • What is the result of the integral after applying the trigonometric substitution?

    -After applying the trigonometric substitution, the integral simplifies to (1/6)dθ, which integrates to (1/6)θ + C.

  • How is the anti-derivative found after back-substitution?

    -After back-substitution, using the relationship θ = arctan(x/6), the anti-derivative is found to be (1/6)(arctan(x/6)) + C.

  • What is the significance of using trigonometric identities in integration?

    -Trigonometric identities are significant in integration as they allow us to transform complex integrals into more manageable forms that can be easily solved, especially when dealing with expressions that involve square terms and constants.

  • What is the final result of the integral after considering the arctangent relationship?

    -The final result of the integral, after considering the arctangent relationship and back-substitution, is (1/6)(arctan(x/6)) + C, where C is the constant of integration.

Outlines
00:00
📚 Solving Integrals with Trigonometric Identities

This paragraph discusses the process of solving a complex indefinite integral using trigonometric identities. The integral in question is 1/(36 + x^2)dx, which initially appears difficult to solve without trigonometry. The speaker suggests using a trig identity that involves 1 + something squared, leading to the rewriting of the integral and the application of the identity 1 + tan^2(theta). The explanation includes a detailed derivation of the derivative of tan(theta) and the subsequent steps to simplify the integral using u-substitution with the trigonometric identity. The paragraph ends with the integral being reduced to a simpler form, setting the stage for the next steps in the solution process.

05:01
🧠 Deriving the Derivative and Simplifying the Integral

In this paragraph, the speaker continues the process of solving the integral by focusing on the derivative of the tangent function. The derivative of tan(theta) is derived using the quotient rule and the basic principles of trigonometry. The resulting expression for the derivative, sec^2(theta), is then used to express the integral in terms of theta. The integral is simplified by canceling out the sec^2(theta) terms, leading to a reduced form of 1/6 d theta. The speaker then explains the back-substitution process to solve for theta, using the arctangent function, and concludes by expressing the final anti-derivative in terms of theta and a constant c. The paragraph provides a clear and comprehensive explanation of the steps involved in simplifying and solving the given integral.

Mindmap
Keywords
💡indefinite integral
The indefinite integral refers to the process of finding a function whose derivative is the given function. In the context of the video, the indefinite integral is the antiderivative of the expression 1 over (36 + x^2). The video discusses the challenge of solving this integral without resorting to trigonometry, which is not straightforward due to the absence of a term like 2x that would allow for u-substitution.
💡trigonometric identities
Trigonometric identities are equations that relate different trigonometric functions such as sine, cosine, and tangent. These identities are fundamental in solving problems in trigonometry and are used extensively in the video to transform the given integral into a more manageable form by relating the expression 1/(36 + x^2) to the tangent squared of an angle.
💡u-substitution
U-substitution is a technique used in calculus to evaluate integrals by replacing the variable of integration with a new variable (u), which simplifies the integral. In the video, the speaker mentions that u-substitution would be easy if there were a 2x term, but the absence of such a term makes the integral more challenging to solve.
💡tangent squared
The term 'tangent squared' refers to the square of the tangent function, which is used in the video to express the denominator of the given integral in terms of trigonometric functions. The transformation of the integral into a form involving tangent squared is crucial for applying trigonometric identities and simplifying the problem.
💡secant squared
Secant squared is the square of the secant function, which is the reciprocal of the cosine function. In the video, secant squared is used to simplify the integral after the expression 1/(36 + x^2) is related to the tangent squared of an angle. This simplification leads to the cancellation of terms and the eventual solution of the integral.
💡arctangent
The arctangent, also known as the inverse tangent function, is used to find the angle whose tangent is a given value. In the video, the arctangent is used to express the variable x in terms of an angle θ, which is crucial for the back substitution process to solve the integral.
💡derivative
A derivative is a fundamental concept in calculus that represents the rate of change of a function with respect to its variable. In the video, the derivative is used to find the relationship between dx and dθ when setting up the trigonometric substitution, which is key to solving the integral.
💡quotient rule
The quotient rule is a formula in calculus used to find the derivative of a quotient of two functions. In the video, the quotient rule is mentioned when calculating the derivative of the tangent function, which is sine(θ)/cosine(θ).
💡unit circle
The unit circle is a circle with a radius of 1 centered at the origin of a coordinate system. It is a fundamental concept in trigonometry as it defines the basic values for sine, cosine, and tangent functions. In the video, the unit circle is implicitly used when discussing the relationship between sine and cosine in the context of the tangent squared identity.
💡back substitution
Back substitution is the process of substituting the values found during the problem-solving process back into the original equation to find the solution. In the video, back substitution is used to express the integral solution in terms of x by substituting the value of θ found from the arctangent function.
💡constant of integration
The constant of integration, often denoted by 'c', is an arbitrary constant that is added to the antiderivative of a function to account for the infinite number of possible antiderivatives that exist for a given derivative. In the video, the constant of integration 'c' is included in the final expression for the indefinite integral.
Highlights

The indefinite integral of 1 over (36 + x^2) is discussed, which is a complex integral without trigonometry.

The speaker expresses difficulty in solving the integral without the presence of a 2x term, which would allow for u-substitution.

The integral is rewritten to emphasize the constant plus something squared, suggesting the use of a trigonometric identity.

The integral is reformulated to resemble 1 + (x^2/36), which is closer to a recognizable trigonometric form.

The trigonometric identity 1 + tan^2(θ) is introduced as a potential solution to simplify the integral.

A proof of the identity 1 + tan^2(θ) = sec^2(θ) is provided, using basic trigonometric relationships.

A substitution is made where x^2/36 is equated to tan^2(θ), leading to a relationship between x and θ.

The derivative of tan(θ) is derived, which is equal to sec^2(θ), using the quotient rule and basic trigonometric derivatives.

The integral is reduced to the simpler form of (1/6)dθ, after canceling out the sec^2(θ) terms.

Back substitution is used to solve for θ, resulting in θ = arctan(x/6).

The anti-derivative of 1/(36 + x^2) is found to be (1/6)θ + C, after back substitution.

The process demonstrates the power of trigonometric identities in simplifying complex integrals.

The method showcases the importance of recognizing patterns and applying appropriate mathematical techniques.

The solution involves a combination of algebraic manipulation and trigonometric transformations.

The speaker's approach to problem-solving is methodical, emphasizing understanding the structure of the problem.

The example serves as a lesson in the application of trigonometric identities and u-substitution in integral calculus.

The final result is a simplified anti-derivative expression, showcasing the effectiveness of the method.

Transcripts

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