Integral of sinx cosx

The Organic Chemistry Tutor
17 Mar 201807:08
EducationalLearning
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TLDRThis video script delves into the integral of sine x cosine x, presenting three distinct methods to obtain the result. The first approach uses u-substitution with u as sine x, leading to one half sine squared x plus a constant. The second method swaps u for cosine x, yielding negative one half cosine squared x plus a constant. The third strategy employs the double angle formula for sine 2x, resulting in negative one fourth cosine 2x plus a constant. The video emphasizes the versatility of the double angle formulas for cosine, demonstrating how different forms can lead to equivalent integral expressions.

Takeaways
  • 📚 The video focuses on finding the integral of sine x multiplied by cosine x using three different methods.
  • 📈 The first method uses u-substitution with u as sine x, leading to the result of (1/2)sin^2(x) + C.
  • 🔄 The second method swaps u with cosine x, yielding -(1/2)cos^2(x) + C after integration.
  • 🔢 The third method involves the double angle formula for sine, specifically sin(2x) = 2sin(x)cos(x), resulting in -(1/4)cos(2x) + C.
  • 🌀 The multiple forms of the double angle formula for cosine 2x are highlighted, which are: cos(2x) = 1 - 2sin^2(x) and cos(2x) = 2cos^2(x) - 1.
  • 🔄 By manipulating the double angle formulas, the integral can be expressed in various forms, all equivalent due to the relationship between the formulas.
  • 📊 The video demonstrates that the integral can be represented in three different but interchangeable ways, showcasing the flexibility in calculus techniques.
  • 🧠 The process of u-substitution and the use of trigonometric identities are key to solving this integral problem.
  • 🎓 The video is an educational resource for understanding integral calculus, particularly in the context of trigonometric functions.
  • 📝 The script emphasizes the importance of understanding and applying the double angle formula and its various forms in solving integrals.
  • 🌟 The video provides a comprehensive look at the problem, offering multiple solutions to enhance the viewer's understanding of calculus concepts.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is finding the integral of sine x cosine x.

  • What are the three techniques discussed in the video for solving the integral?

    -The three techniques discussed are u-substitution, double angle formula of sine 2x, and using the different forms of the double angle formula for cosine 2x.

  • How is u-substitution used in the first method?

    -In the first method, u is substituted with sine x, and du (the derivative of sine) becomes cosine x dx. The integral then becomes the integral of u du, which using the power rule is u to the second power divided by 2 plus a constant C. After substituting back with sine x, the answer is one half sine squared x plus C.

  • What is the second method's approach to finding the integral?

    -The second method makes u equal to cosine x, with du being the derivative of cosine, which is negative sine x dx. The integral then becomes the integral of sine x times u du divided by negative sine x. After simplification, the answer is negative one half cosine squared x plus C.

  • How does the video explain the third method?

    -The third method involves the double angle formula of sine 2x, which is equal to two sine x cosine x. By multiplying both sides by a half, the integral of sine x cosine x is replaced with the integral of one half sine 2x. The antiderivative of sine 2x is negative cosine 2x divided by 2, plus a constant C, resulting in the answer of negative one fourth cosine 2x plus C.

  • Why are there three different answers for the integral of sine x cosine x?

    -There are three different answers because the double angle formula for cosine 2x has multiple forms, and each form can be used to represent the integral of sine x cosine x. The different forms can be converted into each other using the double angle formulas for cosine.

  • What are the two forms of the double angle formula for cosine 2x discussed in the video?

    -The two forms of the double angle formula for cosine 2x discussed are: (1) cosine 2x is equal to 1 minus 2 sine squared x, and (2) cosine 2x is equal to 2 cosine squared x minus 1.

  • How is the first form of the double angle formula for cosine 2x used in the video?

    -The first form, cosine 2x equals 1 minus 2 sine squared x, is used by replacing cosine 2x with this expression and simplifying the integral. The result is positive one half sine squared x plus a constant C.

  • How is the second form of the double angle formula for cosine 2x used in the video?

    -The second form, cosine 2x equals 2 cosine squared x minus 1, is used by replacing cosine 2x with this expression and simplifying the integral. The result is negative one quarter cosine squared x plus one quarter plus a constant C.

  • What is the significance of the different forms of the integral?

    -The different forms of the integral demonstrate the versatility of mathematical approaches and the relationships between different trigonometric identities. They also show how the same integral can be expressed in various ways, offering flexibility in solving problems.

  • How can the different forms of the integral be converted into each other?

    -The different forms of the integral can be converted into each other by using the double angle formulas for cosine. By substituting and simplifying the expressions, one form can be transformed into another, showing their equivalence.

  • What is the final expression for the integral of sine x cosine x after considering all three methods?

    -After considering all three methods, the integral of sine x cosine x can be expressed in any of the three forms: one half sine squared x plus C, negative one half cosine squared x plus C, or negative one fourth cosine 2x plus C. These forms are interchangeable due to the double angle formulas for cosine.

Outlines
00:00
📚 Integral of Sine x and Cosine x - Techniques and Methods

This paragraph introduces the topic of finding the integral of the product of sine x and cosine x. It outlines three distinct methods to derive the answer. The first technique is u-substitution, where u is replaced with sine x, and du becomes cosine x dx. By substituting and applying the power rule, the integral becomes (u^2)/2 + C, which simplifies to (1/2)sin^2(x) + C when substituting back with sine x. The second method involves setting u as cosine x, with du as -sine x dx, leading to the integral of u du/sine x, which simplifies to u^2/2 + C and further to -(1/2)cos^2(x) + C. The third method is based on the double angle formula for sine 2x, which is equal to 2sin x cos x. By substituting this into the integral and simplifying, the result is -(1/4)cos 2x + C. The paragraph concludes by explaining that the three different forms of the integral are due to the multiple forms of the double angle formula for cosine 2x.

05:01
📈 Deriving Integral Forms Using Double Angle Formulas

This paragraph delves into the mathematical process of converting one form of the integral of sine x cosine x into another using the double angle formulas for cosine. It begins by replacing cosine 2x with 1 - 2sin^2x, which results in the integral form of (-1/4)cos 2x + C. The paragraph then proceeds to replace cosine 2x with 2cos^2x - 1, leading to a new integral form. The detailed explanation includes the step-by-step algebraic manipulation and simplification, demonstrating how the different forms of the integral can be derived from one another. This comprehensive summary highlights the versatility and interconnectedness of mathematical formulas in solving integrals.

Mindmap
Keywords
💡Integral
An integral is a mathematical concept that represents the area under a curve or the accumulation of a quantity. In the context of the video, the integral of sine x cosine x is the main focus, and the process of finding it is explained through different techniques. The integral is a fundamental concept in calculus and is used to solve problems involving rates of change and accumulation.
💡Substitution
Substitution is a technique used in calculus to simplify integrals by replacing a complicated expression with a simpler variable. In the video, u-substitution is employed by setting u equal to either sine x or cosine x, which transforms the original integral into a more manageable form. This method is crucial for solving integrals of functions that are products of sine and cosine.
💡Power Rule
The power rule is a fundamental rule in calculus that states if you have an integral of a variable raised to a power, it simplifies to that variable raised to the next power divided by the new exponent. In the video, the power rule is applied when integrating u du, which results in u^2/2 + C, where C is the constant of integration. This rule is essential for calculating integrals of monomials.
💡Derivative
A derivative is a concept in calculus that represents the rate of change of a function with respect to its variable. In the script, the derivatives of sine x (cosine x) and cosine x (-sine x) are used to set up the u-substitution integrals. Understanding derivatives is critical for comprehending how functions change and for solving integrals using techniques like substitution.
💡Double Angle Formula
The double angle formula is a trigonometric identity that expresses a function of a double angle in terms of the function of the single angle. In the video, the double angle formula for sine 2x is used to rewrite sine x cosine x as (1/2)sine 2x, which simplifies the integral. The formula is essential for transforming expressions involving sine and cosine into more manageable forms for integration.
💡Antiderivative
An antiderivative is a function whose derivative is the given function we are trying to integrate. In the context of the video, finding the antiderivative of sine 2x leads to the expression -1/4 cosine 2x + C. The concept of the antiderivative is central to integration, as it represents the reverse process of differentiation.
💡Sine and Cosine
Sine and cosine are fundamental trigonometric functions that relate the angles of a right triangle to the ratios of its sides. In the video, sine x and cosine x are the building blocks of the integral being solved. The relationships between these functions and their derivatives are essential for applying techniques like u-substitution and simplifying integral expressions.
💡Constants
Constants in the context of integration are values that do not change with respect to the variable being integrated. In the video, the constant 'C' represents the arbitrary constant of integration, which is added to the end of each integral solution to account for the infinite number of possible antiderivatives. Constants are a crucial part of the integral's final answer.
💡Rate of Change
Rate of change is a concept in calculus that describes how a quantity changes with respect to another quantity. In the video, the integral of sine x cosine x can be thought of as the accumulation or area under a curve, which is related to the rate of change of the functions sine and cosine. Understanding rates of change is key to solving integrals and applying calculus to real-world problems.
💡Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variables involved. In the video, identities like sine 2x = 2sine x cosine x and the various forms of cosine 2x are used to simplify the integral. These identities are essential for transforming and solving trigonometric integrals.
Highlights

The video focuses on finding the integral of sine x cosine x.

There are three different methods to find the integral.

The first method uses u-substitution with u as sine x.

The derivative of sine is cosine x, leading to du as cosine x dx.

After substitution, the integral becomes u^2/2 + C, with u replaced back with sine x.

The result of the first method is (1/2)sin^2(x) + C.

The second method starts with u as cosine x and du as -sine x dx.

The integral becomes -u^2/2 + C after simplification.

Substituting u back with cosine x, the result is -(1/2)cos^2(x) + C.

The third method involves the double angle formula for sine 2x.

Sine 2x is equal to two sine x cosine x.

The integral is rewritten as (1/2)sin(2x)dx, with the constant moved to the front.

The antiderivative of sine is negative cosine, and the angle is divided by the derivative of the angle (2).

The result from the third method is -(1/4)cos(2x) + C.

Multiple forms of the double angle formula for cosine 2x are discussed.

Cosine 2x can be written as 1 - 2sin^2(x) or 2cos^2(x) - 1.

The integral can be expressed in any of the three forms due to the interchangeable nature of the double angle formulas.

Each form can be converted into another using the double angle formulas for cosine.

The video provides a comprehensive understanding of integrals involving sine and cosine functions.

Transcripts

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Calculus TechniquesIntegral MethodsTrigonometric IdentitiesDouble Angle FormulasU-SubstitutionAntiderivative CalculationMathematical AnalysisSine and CosineEducational ContentMath Tutorial