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Bikram
India
เข้าร่วมเมื่อ 7 ส.ค. 2021
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Integration of √(1 - 9x²) | Advanced Integration Techniques Simplified
*📌 Problem Statement:*
In this video, we solve the integral of the square root of (1 - 9x²) with respect to x. This problem is a great example of using trigonometric substitution to simplify and solve square root expressions.
---
*📋 Steps Explained in the Video:*
1. *Trigonometric Substitution:*
- Let x = (1/3) * sin(theta).
- This transforms the term (1 - 9x²) into cos²(theta).
- Differentiate x to express dx in terms of theta.
2. *Simplification:*
- Substitute x and dx into the integral, reducing it to a trigonometric integral involving cos(theta).
3. *Integration Using Standard Formulas:*
- Solve the simplified integral step by step using basic trigonometric integration techniques.
4. *Back Substitution:*
- Convert the solution back from theta to x using the substitution x = (1/3) * sin(theta).
---
*🔑 Key Insights and Concepts:*
- The substitution x = (1/3) * sin(theta) is critical for simplifying the square root term.
- Learn how to apply trigonometric identities to solve complex integrals.
- A detailed step-by-step explanation ensures clarity for all students.
---
*💡 Conclusion:*
The final solution is presented in a clear and systematic way, making it easy to understand and replicate for similar problems. Watch the video to learn more! 😊
---
In this video, we solve the integral of the square root of (1 - 9x²) with respect to x. This problem is a great example of using trigonometric substitution to simplify and solve square root expressions.
---
*📋 Steps Explained in the Video:*
1. *Trigonometric Substitution:*
- Let x = (1/3) * sin(theta).
- This transforms the term (1 - 9x²) into cos²(theta).
- Differentiate x to express dx in terms of theta.
2. *Simplification:*
- Substitute x and dx into the integral, reducing it to a trigonometric integral involving cos(theta).
3. *Integration Using Standard Formulas:*
- Solve the simplified integral step by step using basic trigonometric integration techniques.
4. *Back Substitution:*
- Convert the solution back from theta to x using the substitution x = (1/3) * sin(theta).
---
*🔑 Key Insights and Concepts:*
- The substitution x = (1/3) * sin(theta) is critical for simplifying the square root term.
- Learn how to apply trigonometric identities to solve complex integrals.
- A detailed step-by-step explanation ensures clarity for all students.
---
*💡 Conclusion:*
The final solution is presented in a clear and systematic way, making it easy to understand and replicate for similar problems. Watch the video to learn more! 😊
---
มุมมอง: 42
วีดีโอ
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Evaluate integral 9cosx-sinx/4sinx+5cosx dx
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Evaluate integral 9cosx-sinx/4sinx 5cosx dx
nice
Thanks 🙏🏻
We must put a positive and negative sign in front of the result of integration.
Okay thanks for correction ☺️
Wao😮
Audio sounds like I’m being abducted by aliens
😂😂😂 it is 40hz Music it's good for brain neurons to connect and engage and also improves focus and concentration
les techniques utilises sont la substitution et l'integration par parties
Yes
Muy buena solución Otra respuesta es: - 2 cosec 2x + C
thanks for your contribution ❤️
I think that if we convert cos³x to cos²x⋆cosx and then replace cos²x with 1-sin²x the integral is solved more easily.
yeah that think also work
Int(cos2x/cos²xsin²x)dx = Int(cos2x/(1/2sin2x)²dx because sin2x = 2cosxsinx therefore cosxsinx=1/2.sin2x = 4.Int(cos2x/sin²2x)dx = 4.1/2.Int(cos2x/sin²2x)d(2x) because d(2x) = 2.dx therefore dx = 1/2.d(2x) = 2.Int(d(sin2x)/sin²2x) because d(sin2x) = cos2x.d(2x) = 2.Int(dt/t²) with t = sin2x = 2.Int(t^-2dt) = 2.t^-1/(-1) + c = -2/t + c = -2/sin2x + c note that the solution in the video is the same: -cotx - tanx = -cosx/sinx - sinx/cosx = -(cos²x + sin²x)/cosxsinx = -1/cosxsinx = -2/2cosxsinx = -2/sin2x
Int(dx/(4+16x²) = 1/4.Int(dx/(1+4x²)) = 1/4.Int(dx/(1+(2x)²)) = 1/4.1/2.Int(d(2x)/(1+(2x)²)) = 1/8.Int(dt/(1+t²)) when t = 2x = 1/8.Arctan(t) + c = 1/8.Arctan(2x) + c
Okay 👌🏻
I did it by considering sinx=t and hence cosxdx=dt and after solving got my answer as (sinx)⁴/4 - (sinx)⁶/6 +C Please someone check whether I have made a mistake or not
yeah it also a way to it
which program do you write with?
OpenBoard
sin x + cos x = sqrt(2)*sin(x+pi/4). So abs(sin x + cos x) = sqrt(2)* abs(sin (x+pi/4)). This function is periodic sin halwave with aplitude sqrt(2) and beginning n -pi/4
Nice
Thanks
2(sinx+cosx)..... May be
Just looking at it , without even playing the video, sin 2X = 2 sin x coz x and sin sq x + coz sq x = 1 . So by the algebraic identity this becomes integral of (sin x + cos x ) . Which is cos x - sin x . One more thing . Integrals must always have limits.
Indefinite integrals dont have limits tho
Good video man, I hope you improve more and make the channel bigger.
thanks brother its make me feel good, thanks a lot
Awesome
Thanks ❤️