TelcoSolve®

# Mathematical Integration.. Can anyone identify the fault here.. :)

∫[-2,2] (x^3 cos((x/2)+(1/2))) * √(4-x^2) dx

Lemme apply integration by parts. Let’s choose:

u = x^3 dv = cos((x/2)+(1/2)) * √(4-x^2) dx

du/dx = 3x^2 v = ∫cos((x/2)+(1/2)) * √(4-x^2) dx

Lets solve v using the substitution u = (x/2)+(1/2):

v = ∫cos(u) * √(4-(2u-1)^2) * 2 du

Let z = 2u-1, so that dz/2 = du:

v = ∫cos((z+1)/2) * √(4-z^2) dz

Let’s evaluate this integral using the substitution z = 2sin(t):

v = ∫cos(sin(t)+1)/2) * √(4-4sin^2(t)) * 2cos(t) dt
= 2 ∫cos(sin(t)+1)/2) * 2cos(t) * 2cos(t) dt
= 8 ∫cos(t)cos(sin(t)/2 + π/4) dt
= 8 ∫cos(t)cos(sin(t)/2)cos(π/4) – sin(t)sin(sin(t)/2)sin(π/4) dt
= 8 ∫cos(t)cos(sin(t)/2)cos(π/4) dt (since sin(π/4) = cos(π/4) = 1/√2 and ∫sin(t)sin(sin(t)/2) dt = 0)
= 4√2 ∫cos(t)cos(sin(t)/2) dt

Assumption: Let w = sin(t)/2, so that dw = cos(t)/2 dt:

v = 8√2 ∫cos(2w) dw
= 8√2 sin(2w)
= 8√2 sin(sin(t)/2)

Therefore:

∫[-2,2] (x^3 cos((x/2)+(1/2))) * √(4-x^2) dx
= uv|[-2,2] – ∫[-2,2] v du/dx dx
= x^3 * 8√2 sin((x/2)+(1/2)) * √(4-x^2)|[-2,2] – ∫[-2,2] 8√2 sin((x/2)+(1/2)) * 3x^2 dx

Evaluating the limits of integration gives:

(-2)^3 * 8√2 sin((-2/2)+(1/2)) * √(4-(-2)^2) – (2)^3 * 8√2 sin((2/2)+(1/2)) * √(4-(2)^2)

= -16√2 sin(3/2) + 16√2 sin(5/2)

2nd integral can be evaluated by substitution of u = (x/2)+(1/2):

∫[-2,2] 8√2 sin((x/2)+(1/2)) * 3x^2 dx
= 24√2 ∫[-2,2] sin(u) * (2u-1)^2 du
= 24√2 ∫[-3/2,5/2] sin(u) * (2u-1)^2 du (since…since (x/2)+(1/2) goes from -3/2 to 5/2 when x goes from -2 to 2.

Now expand (2u-1)^2 to get:

∫[-3/2,5/2] sin(u) * (2u-1)^2 du
= ∫[-3/2,5/2] sin(u) * (4u^2 – 4u + 1) du
= 4 ∫[-3/2,5/2] sin(u) * u^2 du – 4 ∫[-3/2,5/2] sin(u) * u du + ∫[-3/2,5/2] sin(u) du

The third integral is simply -cos(u) evaluated at the limits of integration:

cos(5/2) + cos(3/2)
The first two integrals can be evaluated using integration by parts with:

u = u^2 dv = sin(u) du
du/dx = 1/2 v = -cos(u)

Then:

∫ sin(u) * u^2 du
= -cos(u) * u^2|[-3/2,5/2] + 2 ∫ cos(u) * u du
= cos(5/2) * (25/4) – cos(3/2) * (9/4) + 2 [sin(u) * u – ∫sin(u) du]|[-3/2,5/2]
= (25/4) cos(5/2) – (9/4) cos(3/2) + 2 [sin(u) * u + cos(u)]|[-3/2,5/2]

Finally, plugging in all the evaluated integrals and simplifying gives:

∫[-2,2] (x^3 cos((x/2)+(1/2))) * √(4-x^2) dx
= -16√2 sin(3/2) + 16√2 sin(5/2) + (25/4) cos(5/2) – (9/4) cos(3/2) + 4√2

Therefore, the value of the integral is approximately 8.564

Cheers!
Shiva

Scroll to Top

## Our History

The employees here at TelcoSolve™ are from early days of Modern Communication Technology / GSM / Wireless Mobile Communication / Analog to Digital Transformation of Services, that grew from ground level work. Could it be with Radio Network Management, OR Could it be Centrally Organized Network & Service Monitoring Center (NOC/SOC) activities.. our employees have lived, loved and leveraged all their  learnings from entire journey in Communication Tech’ space, which resulted in bringing this company up, with their strong  passion instilled!

As our Vision statement represents: We would like to collaborate and partner with likeminded professionals, institutions, organizations, and representative bodies alike to meet requirements of Communication Tech’ services world. And at same time, to help every being on Planet to live with dignity & respect, irrespective of any differences, since, we live on an inclusive planet that hosts diversity of life!

## Our Vision

Telecommunication & Communication Tech’ Services industry segments have been lifeline for the new millennia. Could it be in reducing corruption ..to connecting each and every individual on the Planet, ..to connecting the hybrid platforms, systems, elements and/or devices that cut-cross several industry segments in making intelligent, logical, ethical & lawful decisions on the actionable data/reports.

Here at TelcoSolve™, we would like to collaborate and partner with likeminded professionals, institutions, organizations, and representative bodies alike to meet requirements of Communication Tech’ services world. And at same time, to help every being on Planet to live with dignity & respect, irrespective of any differences, since, we live on an inclusive planet that hosts diversity of life!

## Our Workplace Policy:

TelcoSolve® has updated its workplace policy effective January 2023, which includes the following provisions:

1. Currently, TelcoSolve operates under Hybrid Work Norms and has seating arrangements at NASSCOM®’s facilities in Bangalore City, India, supported by the Karnataka State Government.
2. For client-facing projects, a mutual workplace policy agreement will be signed between TelcoSolve and its clients before any project assignments commence. This agreement will also cover global compliances such as IFRS15, IR35, OHS, GST/iGST, LUT, IEC, etc. based on the business requirements of the project.
3. For projects requiring onsite presence of TelcoSolve employees, they will abide by the workplace policy of the client’s office in that geography.
4. Incentives for both Fixed-Term and Rolling Contracts will be mutually agreed and accepted between TelcoSolve management and its employees before project assignments commence.
5. Regional and country-specific vaccination policies will be in place in mutual agreement with clients and commercial partners.
6. For reverse engineering services, such engagements will fully comply with the region and country-specific trade laws and client contracts, including working on outdated systems and end-of-service products.