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∫ [arg|z|] dx from 0 to 100 with z ≠ 0 | JEE

JEE Maths question with a full step-by-step solution.

Question
If z≠0z\ne 0, then ∫x=0100[arg⁡∣z∣] dx\displaystyle\int_{x=0}^{100}\big[\arg|z|\big]\,dx, where [ ⋅ ][\,\cdot\,] denotes the greatest integer function, is
A00correct
B1010
C100100
Dnot defined
Solution
Step 1: Examine ∣z∣|z|. For any non-zero complex number zz, the modulus ∣z∣|z| is a positive real number, lying on the positive real axis of the complex plane. Step 2: A positive real number has argument 00, so
arg⁡∣z∣=0.\arg|z|=0.
 Step 3: Apply the greatest integer function: [arg⁡∣z∣]=[0]=0\big[\arg|z|\big]=[0]=0. This is a constant, independent of the integration variable xx. Step 4: Integrate the constant zero:
∫01000 dx=0.\int_{0}^{100}0\,dx=0.
 Correct answer: (1)
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