the integration 22

Calculus Level 2

∫ π 2 2 π ⌈ cos ⁡ x 2 ⌉ x d x = ? \large \int_\frac \pi 2^{2\pi} \frac {\left \lceil \cos \frac x2 \right \rceil}x dx = \ ?


The answer is 0.6931.

Aly Ahmed by Aly Ahmed , 34, Egypt

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2 solutions

Chew-Seong Cheong
Jun 29, 2020

I = ∫ π 2 2 π ⌈ cos ⁡ x 2 ⌉ x d x = ∫ π 2 π ⌈ cos ⁡ x 2 ⌉ x d x + ∫ π 2 π ⌈ cos ⁡ x 2 ⌉ x d x = ∫ π 2 π 1 x d x + ∫ π 2 π 0 x d x = ln ⁡ x ∣ π 2 π = ln ⁡ π π 2 = ln ⁡ 2 ≈ 0.693 \begin{aligned} I & = \int_\frac \pi 2^{2\pi} \frac {\left \lceil \cos \frac x 2 \right \rceil}x dx \\ & = \int_\frac \pi 2^\pi \frac {\left \lceil \cos \frac x 2 \right \rceil}x dx + \int_\pi^{2\pi} \frac {\left \lceil \cos \frac x 2 \right \rceil}x dx \\ & = \int_\frac \pi 2^\pi \frac 1x dx + \int_\pi^{2\pi} \frac 0x dx \\ & = \ln x \bigg|_\frac \pi 2^\pi = \ln \frac \pi {\frac \pi 2} = \ln 2 \approx \boxed{0.693} \end{aligned}

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@Chew-Seong Cheong it should be cos(x/2), not cos(pi/2).

Chiang Jun Siang - 11 months, 2 weeks ago

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Thanks. I have changed it.

Chew-Seong Cheong - 11 months, 2 weeks ago

The value of the integral is ln ⁡ π − ln ⁡ ( π 2 ) = ln ⁡ 2 ≈ 0.6931 \ln π-\ln (\frac{π}{2})=\ln 2\approx \boxed {0.6931} .

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