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Integral Calculus> integration from 0 to pi |1/2 + cos x | d...

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integration from 0 to pi |1/2 + cos x | dx
can someone please help me with this

shiya , 6 Years ago

Grade 12th pass

1 Answers

Samyak Jain

Last Activity: 6 Years ago

Examine the graph of 1/2 + cosx from x=0 to $\pi$ , you will find that it is negative for x $\small \epsilon$ (2 $\small \pi$ /3 , $\small \pi$ ]

$\small \therefore$ $\small \int_{0}^{\pi}$ |1/2 + cosx| = $\small \int_{0}^{2\pi /3}$ (1/2 + cosx) – $\small \int_{2\pi /3}^{\pi}$ (1/2 + cosx)

= $\small \[x/2 + sinx]_{0}^{2\pi /3}$ – $\small \[x/2 + sinx]_{2\pi /3}^{\pi}$

= [( $\small \pi$ /3 + sin2 $\small \pi$ /3) – (0 + 0)] – [( $\small \pi$ /2 + 0) – ( $\small \pi$ /3 + sin2 $\small \pi$ /3)]

= ( $\small \pi$ /3 + $\small \sqrt{3}$ / 2) – $\small \pi$ /2 + ( $\small \pi$ /3 + $\small \sqrt{3}$ / 2) = 2 $\small \pi$ /3 – $\small \pi$ /2 + $\small \sqrt{3}$

= $\small \pi$ /6 + $\small \sqrt{3}$

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