How do I prove S i ( x ) = π 2 + O ( x − 1 ) � � ( � ) = � 2 + � ( � − 1 ) S i ( x ) = ∫ x 0 s i n t t d t � � ( � ) = ∫ 0 � � � � � � � � ?

 Unfortunately, the integral expression provided is incomplete and lacks an integrand s(t). To prove the statement:

S_i(x) = π/2 + O(x - 1)

where S_i(x) represents the integral:

S_i(x) = ∫_0^x sin(t) / t dt

you need to specify the definition of s(t) and perform the integration with proper handling of the singularity at t=0. Once you provide the complete definition of s(t), I can assist you with the analysis and proof.

Here's a breakdown of the steps involved:

1. Define s(t): What function does s(t) represent? Explain its relationship to sin(t) / t.
2. Integrate: Perform the integration:

S_i(x) = ∫_0^x s(t) dt

3. Analyze the singularity: How do you handle the limit as t approaches 0? This is crucial for analyzing the asymptotic behavior and proving the O-notation statement.
4. Prove the O-notation: Show that the error term in the approximation S_i(x) ≈ π/2 is indeed O(x - 1) as x approaches 1.

Remember, providing the complete information about s(t) is essential for the analysis and proof.