Reduction Formulas For Integration

The Organic Chemistry Tutor
23 Mar 201812:26
EducationalLearning
32 Likes 10 Comments

TLDRThe video script presents a comprehensive guide on finding the indefinite integral of trigonometric functions using reduction formulas. It demonstrates the step-by-step process of integrating sine squared, cosine cubed, and sine to the fourth power of x. The explanation includes applying the reduction formula, simplifying the result, and utilizing double angle formulas to achieve the final answer in a clear and engaging manner.

Takeaways
  • 📚 The indefinite integral of sine squared x (∫sin^2(x) dx) can be found using the reduction formula for sine.
  • 🔄 When n=2, the formula simplifies to -1/2 cos(x)sin(x) + 1/2 ∫sin^0 dx, which results in 1/2 x after integrating sin^0 (which is 1).
  • 📈 The final answer for the integral of sine squared x is 1/2 x + C, where C is the constant of integration.
  • 🌟 The double angle formula for sine (sin(2x) = 2sin(x)cos(x)) is used to further simplify the expression to -1/4 sin(2x) + C.
  • 🧩 The indefinite integral of cosine cubed x (∫cos^3(x) dx) is also calculated using the reduction formula for cosine.
  • 🔢 For cosine cubed, with n=3, the integral becomes (1/3)cos^2(x)sin(x) + (2/3)∫cos(x)dx, after applying the formula.
  • 🏹 By replacing cos^2(x) with (1 - sin^2(x)), the integral simplifies to (1/3)sin(x) - (1/3)sin^3(x) + C.
  • 🔄 The reduction formula for sine is key in simplifying higher power sine integrals, such as ∫sin^4(x) dx.
  • 📊 With n=4, the integral of sine to the fourth power becomes -1/16 sin(2x) - 3/16 sin(4x) + C after applying the formula and simplifying.
  • 🔧 The power reducing formula for sine squared is used to transform sin^4(x) into a form involving sin(2x) and sin(4x), aiding in the simplification process.
  • 📌 The process of integrating trigonometric functions using reduction formulas and double angle formulas is demonstrated through several examples, providing a clear method for tackling similar problems.
Q & A
  • What is the reduction formula for sine?

    -The reduction formula for sine states that the integral of sine raised to the power of n, denoted as sin^n(x) dx, is equal to -1/n * cos(x) * sin^(n-1)(x) + (n-1)/n * integral of sin^(n-2)(x) dx.

  • How does the reduction formula apply to the integral of sine squared, sin^2(x) dx?

    -For the integral of sine squared, n equals 2. Applying the reduction formula, we get -1/2 * cos(x) * sin(x) + 1/2 * integral of sin^0(x) dx. Since anything raised to the power of 0 is 1, the integral simplifies to -1/2 * cos(x) * sin(x) + 1/2 * x, plus the constant of integration, C.

  • What is the final form of the indefinite integral of sine squared, sin^2(x) dx, using the double angle formula for sine?

    -Using the double angle formula for sine, the final form of the indefinite integral of sine squared is -1/4 * sin(2x) + 1/2 * x + C.

  • What is the reduction formula for cosine?

    -The reduction formula for cosine states that the integral of cosine raised to the power of n, denoted as cos^n(x) dx, is equal to 1/n * cos^(n-1)(x) * sin(x) + (n-1)/n * integral of cos^(n-2)(x) dx.

  • How does the reduction formula apply to the integral of cosine cubed, cos^3(x) dx?

    -For the integral of cosine cubed, n equals 3. Applying the reduction formula, we get 1/3 * cos^2(x) * sin(x) + 2/3 * integral of cos^1(x) dx. Since the integral of cos(x) is sin(x), the expression simplifies to sin(x) - 1/3 * sin^3(x) + C.

  • How can we rewrite the expression for the integral of cosine squared, cos^2(x) dx, using the identity cos^2(x) = 1 - sin^2(x)?

    -By using the identity cos^2(x) = 1 - sin^2(x), we can rewrite the expression for the integral of cosine squared as 1/3 * sin(x) - 1/3 * sin^3(x) + C.

  • What is the integral of sine to the fourth power, sin^4(x) dx, using the reduction formula for sine?

    -Applying the reduction formula for sine with n equal to 4, we get -1/4 * cos(x) * sin^3(x) + 3/4 * integral of sin^2(x) dx. After distributing and simplifying, the final answer is 3/8 * x - 1/4 * sin^2(x) + 1/32 * sin^4(x) + C.

  • How can the double angle formula for sine be used to simplify the expression for the integral of sine to the fourth power?

    -By using the double angle formula for sine, sin(2x) = 2 * sin(x) * cos(x), we can rewrite the term -1/4 * sin^2(x) as -1/8 * sin(2x). This simplifies the expression to 3/8 * x - 1/8 * sin(2x) + 1/32 * sin^4(x) + C.

  • What is the power reducing formula for sine squared?

    -The power reducing formula for sine squared is sin^2(x) = 1/2 * (1 - cos(2x)). This formula allows us to express sine squared in terms of a cosine function and a constant.

  • How can the power reducing formula for sine squared be applied to the expression for the integral of sine to the fourth power?

    -By applying the power reducing formula for sine squared, we can replace sin^2(x) with 1/2 * (1 - cos(2x)) in the expression for the integral of sine to the fourth power. This results in the term -1/16 * sin(2x) * (1 - cos(2x)) which further simplifies to -1/16 * sin(2x) + 1/16 * sin(4x), plus other terms.

  • What is the final simplified form of the indefinite integral of sine to the fourth power, sin^4(x) dx?

    -After simplifying using the double angle and power reducing formulas, the final form of the indefinite integral of sine to the fourth power is 3/8 * x - 1/4 * sin^2(x) + 1/32 * sin^4(x) + C.

  • What is the role of the constant of integration, C, in the indefinite integrals found using reduction formulas?

    -The constant of integration, C, is added to the indefinite integrals to account for the arbitrary constant that arises from the indefinite nature of the integration process. It represents the family of functions that are solutions to the differential equation.

Outlines
00:00
📚 Finding the Indefinite Integral of sin^2(x) using Reduction Formula

This paragraph explains the process of finding the indefinite integral of sine squared of x (sin^2(x)) using the reduction formula for sine. It starts by stating the formula for the integral of sine raised to the power of n, which is -(1/n) * (cos(x) * sin^(n-1)(x) + (n-1)/n * ∫sin^(n-2)(x) dx). The video then applies this formula with n=2, leading to the result of (1/2) * x + (1/2) * ∫sin^0(x) dx, which simplifies to (1/2) * x + C. The speaker also introduces the double angle formula for sine, sin(2x) = 2 * sin(x) * cos(x), to further simplify the result into a form involving sin(2x), resulting in the final answer of (1/2) * x - (1/4) * sin(2x) + C.

05:01
📝 Solving the Indefinite Integral of cos^3(x) using Reduction Formula

This paragraph focuses on finding the indefinite integral of the cosine cubed of x (cos^3(x)) using the reduction formula for cosine. The formula for the integral of cosine raised to the power of n is given as (1/n) * (cos^(n-1)(x) * sin(x) + (n-1)/n * ∫cos^(n-2)(x) dx). With n=3, the paragraph calculates the integral to be (1/3) * cos^2(x) * sin(x) + (2/3) * ∫cos^(-2)(x) dx. It then replaces cos^2(x) with (1 - sin^2(x)) and proceeds to distribute (1/3) * sin(x) to both terms, resulting in the final answer of sin(x) - (1/3) * sin^3(x) + C.

10:03
🔢 Integrating sin^4(x) using Reduction and Double Angle Formulas

The paragraph details the integration of sine to the fourth power of x (sin^4(x)) using the reduction formula for sine and the double angle formulas. It begins by applying the formula for the integral of sine raised to the power of n, resulting in (-1/4) * cos(x) * sin^3(x) + (3/4) * ∫sin^2(x) dx. The speaker then uses the previously derived result of the integral of sine squared (1/2 * x - 1/4 * sin(2x) + C) and applies the double angle formulas for sine and cosine to simplify the expression. The final answer is presented in two forms: one involving sin(2x) and another involving sin(4x), which is (3/8) * x - (1/4) * sin(2x) - (1/32) * sin(4x) + C.

Mindmap
Keywords
💡indefinite integral
The indefinite integral represents a family of functions that differentiate to the same function. In the context of the video, it is used to find the antiderivative of a given function, such as sine squared x or cosine cubed x. The process involves using reduction formulas for sine and cosine to simplify the integral into a more manageable form.
💡reduction formula
A reduction formula is a mathematical formula used to simplify the process of integration by reducing the power of the function being integrated. In the video, the reduction formulas for sine and cosine are applied to find the indefinite integrals of sine squared x, cosine cubed x, and sine to the fourth x.
💡sine
Sine is a trigonometric function that relates the ratio of the opposite side to the hypotenuse in a right triangle to the angles of a unit circle. In the video, the integration of sine-based functions is the primary focus, with the use of reduction formulas and double angle formulas to simplify the integral expressions.
💡cosine
Cosine is a trigonometric function similar to sine, but it represents the ratio of the adjacent side to the hypotenuse in a right triangle or the x-coordinate on the unit circle. The video discusses the integration of cosine-based functions, particularly using reduction formulas to find their indefinite integrals.
💡double angle formula
The double angle formula is a trigonometric identity that expresses a trigonometric function of double an angle in terms of the function of the single angle. For sine, it states that sine 2x is equal to two times sine x times cosine x. This formula is used in the video to further simplify the expressions obtained from the reduction formulas.
💡power reducing formula
The power reducing formula is a trigonometric identity that expresses a trigonometric function of a multiple angle in terms of functions of the single angle. For sine squared, it states that sine squared x is equal to one half minus cosine 2x. This formula is used in the video to simplify the integration of sine to higher powers.
💡integration
Integration is the process of finding the antiderivative, or the function whose derivative equals a given function. In the video, integration is the main mathematical operation, with a focus on integrating trigonometric functions like sine and cosine using various formulas and techniques.
💡antiderivative
An antiderivative is a function whose derivative is equal to the given function. In the context of the video, finding the antiderivative of a function like sine squared x or cosine cubed x is the goal, and it is achieved through the application of integration techniques and formulas.
💡trigonometric functions
Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratios of its sides or the coordinates of points on a unit circle. In the video, sine and cosine are the primary trigonometric functions used in the integration problems.
💡trigonometric identities
Trigonometric identities are equations that hold true for all values of the angles involved. They are used to simplify and manipulate trigonometric expressions. In the video, identities like the double angle formula and the power reducing formula for sine are crucial for simplifying the integration process.
Highlights

The use of the reduction formula for sine to find the indefinite integral of sine squared of x.

Setting n to 2 in the reduction formula to match the problem's requirement.

Deriving the indefinite integral of sine squared x as (1/2)x + C using the reduction formula.

The application of the double angle formula for sine in adjusting the final answer.

Transforming the indefinite integral of sine squared x to involve sine of 2x for a different representation.

The use of the reduction formula for cosine to find the indefinite integral of cosine cubed x.

Setting n to 3 in the cosine reduction formula for the specific problem.

Deriving the indefinite integral of cosine cubed x as sine x - (1/3)sine^3 x + C.

Substituting cosine squared with (1 - sine squared) for further simplification.

Using the power reducing formula for sine squared to express sine^3 x in terms of sine and cosine.

Integrating sine to the fourth power x using the reduction formula for sine.

Setting n to 4 in the sine reduction formula for the given problem.

Applying the previously found indefinite integral of sine squared x in the current problem.

Adjusting the final answer using the double angle formula for sine and the power reducing formula for sine squared.

Rewriting sine^3 x as sine x times sine squared and applying the double angle formula for cosine.

Combining like terms and simplifying the expression to get the final answer in a different form.

The final answer for the indefinite integral of sine to the fourth power x is expressed in terms of sine and cosine of higher angles.

The process demonstrates the versatility of trigonometric identities and reduction formulas in solving integrals.

Transcripts
Rate This

5.0 / 5 (0 votes)

Thanks for rating: