MHB What's the best strategy to solving this Integral in 3 minutes?

  • Thread starter Thread starter Lorena_Santoro
  • Start date Start date
  • Tags Tags
    Integral Strategy
AI Thread Summary
To solve the integral of cos^3(2x), it is effective to separate it into (cos^2(2x))cos(2x) and use the identity cos^2(2x) = 1 - sin^2(2x). A substitution of u = sin(2x) simplifies the integral, leading to du = 2cos(2x)dx, which allows for the transformation of cos(2x)dx into (1/2)du. The integral then evaluates to (1/2)∫(1 - u^2)du from 0 to sin(8), resulting in the expression sin(8) - (sin^3(8)/3). This method demonstrates a clear strategy to solve the integral efficiently within a limited time frame.
Lorena_Santoro
Messages
22
Reaction score
0
 
Mathematics news on Phys.org
Please do not double post here, either.

-Dan
 
Separate cos^3(2x) as (cos^2(2x))cos(2x)= (1- sin^2(2x))cos(2x). Now use the substitution u= sin(2x) so that du= 2cos(2x)dx, cos(2x)dx= (1/2)du. When x= 0, u= sin(0)= 0 and when x= 4, u= sin(8). The integral becomes $$\frac{1}{2}\int_0^{sin(8)}(1- u^2)du=\left[u- \frac{u^3}{3}\right]_0^{sin(8)}$$$$= sin(8)- \frac{sin^3(8)}{3}$$.
 
$x = \dfrac{\pi}{4} \implies \text{ upper limit }, u= 1$
 
Last edited by a moderator:
Yes, and thank you!
 

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
View full post »

Thread 'Fermat's Last Theorem'

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Similar threads

I What is the Best Strategy for the 100 Prisoner Problem?
Replies
6
Views
2K
MHB Betting Strategy: Win Profits 90% of the Time
Replies
4
Views
1K
I Why integrals took 2000 years to come up in a rigorous manner?
Replies
9
Views
2K
Studying Advice for How to Attack Problems
Replies
24
Views
2K
B Strategies for Solving Mathematical 'Riddles' in Scholarship Tests
Replies
3
Views
1K
MHB Exponential Equation solve 54⋅2^(2x)=72^x⋅√0.5
Replies
3
Views
2K
Help finding winning strategies in the following tile game?
Replies
2
Views
2K
I Integrating a product of exponential and trigonometric functions
Replies
21
Views
3K
MHB Find the amount of sand in each bag
Replies
1
Views
1K
MHB Is There a Strategy for Finding Primitive Roots of 2p^n If There is One for p^n?
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top