Evaluate the definite integral in the given problem

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The discussion focuses on evaluating a definite integral using integration by parts. The participants agree that integration by parts appears to be the most straightforward method for solving the problem. Key variables are identified, including u=t and dv=sin(1/2 t). There is uncertainty about whether alternative integration methods could be more effective. Ultimately, integration by parts is suggested as the primary approach for this integral.
chwala
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Homework Statement
See attached.
Relevant Equations
Integration by parts.
My interest is on the highlighted part only. Find the problem and solution here.

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1651270902092.png


This is clear to me (easy )...i am seeking an alternative way of integrating this...or can we say that integration by parts is the most straightforward way?

The key on solving this using integration by parts is to note that;
##u=t, du=dt, dv=sin \frac {1}{2} t, v=-2 cos \frac {1}{2} t##
 
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chwala said:
Homework Statement:: See attached.
Relevant Equations:: Integration by parts.

can we say that integration by parts is the most straightforward way?
I am not sure but afraid that we have no better way.
 
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Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
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