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AS & A Level · AS/A Level

Further Mathematics

Advanced integration and improper integrals

Name: ____________________Date: October 10, 2026
  1. 1.

    Find integral ln x dx for x > 0 using integration by parts.

    [3 marks] · no calculator
  2. 2.

    Evaluate integral from 0 to 1 of x/(1 + x^2) dx exactly.

    [3 marks] · no calculator
  3. 3.

    Determine for which real p the improper integral from 1 to infinity of x^(-p) dx converges, and give its value when it does.

    [3 marks] · no calculator
  4. 4.

    Evaluate integral from 0 to 1 of x^2 e^x dx exactly.

    [3 marks] · no calculator
  5. 5.

    Let I_n = integral from 0 to pi/2 of sin^n x dx. For n >= 2 derive I_n = (n - 1)I_(n - 2)/n, and find I_4 exactly.

    [3 marks] · no calculator
  6. 6.

    A learner claims integral from -1 to 1 of 1/x^2 dx equals -2 by using -1/x at the endpoints. Explain the error and determine convergence.

    [3 marks] · no calculator

Original, unofficial practice questions from SubjectScout — not official past-paper questions.

Marking points are indicative, not an official mark scheme. Accept equivalent valid methods and supported interpretations that address the task; award each mark once without requiring the model wording.