Reduction Formulas For Integration
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
📚 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.
📝 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.
🔢 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
💡reduction formula
💡sine
💡cosine
💡double angle formula
💡power reducing formula
💡integration
💡antiderivative
💡trigonometric functions
💡trigonometric identities
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
Browse More Related Video
Indefinite Integral
Simplifying Trigonometric Expressions Using Power Reducing Formulas
Integral of sinx cosx
How to Use the Double and Half Angle Formulas for Trigonometry (Precalculus - Trigonometry 28)
Inverse Trig Functions With Double Angle Formulas and Half Angle Identities - Trigonometry
Formulas for Trigonometric Functions: Sum/Difference, Double/Half-Angle, Prod-to-Sum/Sum-to-Prod
5.0 / 5 (0 votes)
Thanks for rating: