Again a maximum problem...

Calculus Level 5

∫ 0 2 n π m a x { sin ⁡ x , arcsin ⁡ ( sin ⁡ x ) } d x \int_{0}^{2n\pi} max\{\sin x,\arcsin(\sin x)\} dx can be expressed as n × π a − b c n \times \frac{\pi^{a} - b}{c} . What is the value of a + b + c \displaystyle a+b+c ?

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The answer is 14.

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Abhishek Singh by Abhishek Singh , 24, India

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1 solution

Abhishek Singh
Oct 10, 2014

This integral can be divided as = n [ ∫ 0 π 2 x d x + ∫ π 2 π ( π − x ) d x + ∫ π 2 π sin ⁡ x d x ] =n\Bigg[ \int_{0}^{\frac{\pi}{2}}x dx +\int_{\frac{\pi}{2}}^{\pi} (\pi - x) dx + \int_{\pi}^{2 \pi} \sin x dx\Bigg] = n [ π 2 8 + π 2 2 − 1 2 × ( π − π 2 4 ) − 2 ] = n\Bigg[\frac{\pi^{2}}{8}+\frac{\pi^{2}}{2}-\frac{1}{2} \times \Bigg(\pi-\frac{\pi^{2}}{4}\Bigg) - 2\Bigg] Solving this we get I = n × π 2 − 8 4 \displaystyle I= \boxed{ n \times \frac{\pi^{2} - 8}{4}}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Dude, sin(arcsin x) = x. Check your question,you should have given arcsin (sin x) instead of sin(arcsin x)

Ayush Garg - 6 years, 7 months ago

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I've changed it sorry for the mistake

Abhishek Singh - 6 years, 7 months ago

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