MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions
Problem
Suppose H is an antiderivative of h(x)=x2+x2x+1 defined for all x>−1.
You are told that H(0)=3 and that H(1)=2.
(a) Compute the most general form of H(x) by evaluating ∫h(x)dx. Show each step of your integration, including algebraic manipulation where needed.
(b) Use the condition H(0)=3 to determine the value of the integration constant, and write the resulting particular antiderivative.
(c) Evaluate your result from part (b) at x=1. Compare the computed value with the second given condition H(1)=2, and state whether the two conditions are simultaneously consistent.
(d) Provide a clear mathematical justification for your conclusion in part (c). If the conditions are inconsistent, explain precisely why no single antiderivative of h can satisfy both points; if they are consistent, confirm that your antiderivative meets both conditions.