<p>Evaluate: \(\displaystyle\int_0^{5\pi/12} [\tan x]\,dx\), where [·] is the greatest integer function (up to four decimal places).</p>
Step-by-Step Solution
Key Concept: Split the integral at points where tan(x) crosses integer values. Since tan(x) is increasing on [0, 5π/12], identify where tan(x) = 1 and tan(x) = 2, then apply the GIF definition: ∫[tan x]dx = ∫0·dx + ∫1·dx + ∫2·dx over appropriate subintervals.
<p><strong>Step 1:</strong> Determine the range of tan(x) on [0, 5π/12]. At x = 0, tan(0) = 0. At x = 5π/12, tan(5π/12) = tan(75°) = 2 + √3 ≈ 3.732.</p><p><strong>Step 2:</strong> Find breakpoints where tan(x) equals integers:<br/>• tan(x) = 1 at x = π/4 ≈ 0.7854<br/>• tan(x) = 2 at x = arctan(2) ≈ 1.1071<br/>• tan(x) = 3 at x = arctan(3) ≈ 1.2490</p><p><strong>Step 3:</strong> Split the integral:<br/>∫₀^(5π/12) [tan x]dx = ∫₀^(π/4) 0·dx + ∫_(π/4)^(arctan 2) 1·dx + ∫_(arctan 2)^(arctan 3) 2·dx + ∫_(arctan 3)^(5π/12) 3·dx</p><p><strong>Step 4:</strong> Evaluate each piece:<br/>• First integral = 0<br/>• Second integral = arctan(2) - π/4 ≈ 1.1071 - 0.7854 = 0.3217<br/>• Third integral = 2[arctan(3) - arctan(2)] ≈ 2(1.2490 - 1.1071) = 0.2838<br/>• Fourth integral = 3[5π/12 - arctan(3)] ≈ 3(1.3090 - 1.2490) = 0.1800</p><p><strong>Step 5:</strong> Sum all parts: 0 + 0.3217 + 0.2838 + 0.1800 = 0.7855</p><p>∴ Answer: <strong>0.7855</strong></p>
Correct Answer: 0