Definite Integration Question - None

Question

Let $ I = \int_0^{5\pi/12} [\tan x] \, dx $, where $[\cdot]$ denotes the greatest integer function. Find the value of $I$.

Accepted Answer

Step 1: Determine the range of tan x

At x = 0: tan(0) = 0

At x = 5π/12: tan(5π/12) = tan(75°) = 2 + √3 ≈ 3.732

So tan x ∈ [0, 2+√3] on [0, 5π/12]

Step 2: Find critical points where tan x equals integers

[tan x] changes value when tan x crosses integer values 1, 2, and 3.

Let α = tan⁻¹(1) = π/4

Let β = tan⁻¹(2) ≈ 1.1071 rad

Let γ = tan⁻¹(3) ≈ 1.2490 rad

Note: 5π/12 ≈ 1.3090 rad

Step 3: Partition the integration domain

• On [0, π/4): [tan x] = 0, length = π/4

• On [π/4, tan⁻¹(2)): [tan x] = 1, length = tan⁻¹(2) - π/4

• On [tan⁻¹(2), tan⁻¹(3)): [tan x] = 2, length = tan⁻¹(3) - tan⁻¹(2)

• On [tan⁻¹(3), 5π/12]: [tan x] = 3, length = 5π/12 - tan⁻¹(3)

Step 4: Calculate the integral

I = 0·(π/4) + 1·(tan⁻¹(2) - π/4) + 2·(tan⁻¹(3) - tan⁻¹(2)) + 3·(5π/12 - tan⁻¹(3))

I = tan⁻¹(2) - π/4 + 2tan⁻¹(3) - 2tan⁻¹(2) + 5π/4 - 3tan⁻¹(3)

I = -tan⁻¹(2) - tan⁻¹(3) + π

Step 5: Compute numerical value

tan⁻¹(2) ≈ 1.10714871779409

tan⁻¹(3) ≈ 1.24904577239825

I = π - 1.10714871779409 - 1.24904577239825

I = 3.14159265358979 - 2.35619449019234

I ≈ 0.78539816339745

I ≈ 0.7854 (to 4 decimal places)

∴ Answer: 0.7857

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