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...Someverychallengingcalculusproblems joseph breen here are two difcult calculus problems solved using only sophisticated and clever applications of elementary in particular there is no complex analysis or use the residue theorem fourier series anything like that both were basis for talks given at ucla gso seminar integral concatenation integrals from innite was originally evaluated byothermethodsin andthesolutionpresentedbelowisinspiredbythesolutionfrom togetherwith other computations found on internet my own computational decisions contents areallyhardintegral areallyhardinniteseries theintegral we will evaluate z i arccos cosx dx step rewrite integrand with trigonometry then introduce a double webeginwithsometrigonometry recall angle identity cos this implies q letting yields thus r usingthis becomes next consider right triangle picture arctan andso goal aforementioned again make substitution x y thendx dyandwehave cosy dy s p sin b c dt bt t dtdy siny yt adjust constants order to sim...