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Question: The value of \(\int_{0}^{1}{}\)\|sin 2 π x\| dx is equal to...

The value of ∫01\int_{0}^{1}{}|sin 2 π x| dx is equal to

A

0

B

2/π

C

1/π

D

2

Answer

2/π

Explanation

Solution

Since |sin 2 π x | is periodic with period12\frac{1}{2},

I = ∫01\int_{0}^{1}{}|sin 2 π x| dx = 2 ∫01/2∫sin⁡\int_{0}^{1/2\int}\sin2πx dx = 2[−cos⁡2πx2π]012\left\lbrack - \frac{\cos 2\pi x}{2\pi} \right\rbrack_{0}^{\frac{1}{2}}

= 2π\frac{2}{\pi}