- #1
DaalChawal
- 85
- 0
How Large Can This Integral Be?
- Context: MHB
- Thread starter DaalChawal
- Start date
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SUMMARY
The integral $\displaystyle\left|\int_0^\pi\frac{\cos(2017\theta)}{5-4\cos\theta}d\theta\right|$ can be estimated using the properties of the cosine function and the bounds established by $|\cos(2017\theta)| \leq 1$ and $|5 - 4\cos\theta| \geq 1$. By applying these inequalities, the maximum value of the integral is determined to be less than or equal to $\pi$. The analysis leads to the conclusion that the most likely correct choice among the provided options is the one that aligns with this upper bound.
PREREQUISITES- Understanding of integral calculus
- Familiarity with trigonometric functions
- Knowledge of inequalities and their applications in calculus
- Experience with evaluating definite integrals
- Study the properties of trigonometric integrals
- Learn about the application of inequalities in integral estimation
- Explore advanced techniques in integral calculus, such as residue theory
- Investigate the behavior of oscillatory integrals
Mathematicians, calculus students, and anyone interested in advanced integral estimation techniques.
- #2
Opalg
Gold Member
MHB
- 2,778
- 13
$|\cos(2017\theta)|\leqslant1$.
$|5-4\cos\theta|\geqslant1$.
Use those facts to estimate how large $\displaystyle\left|\int_0^\pi\frac{\cos(2017\theta)}{5-4\cos\theta}d\theta\right|$ can be. Then decide which of the four given choices seems most likely to be correct.
$|5-4\cos\theta|\geqslant1$.
Use those facts to estimate how large $\displaystyle\left|\int_0^\pi\frac{\cos(2017\theta)}{5-4\cos\theta}d\theta\right|$ can be. Then decide which of the four given choices seems most likely to be correct.
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