IEMJEE Maths preparation: Integration by substitution — practice worksheet with answers

How to use this worksheet

This independent worksheet checks integration by substitution with a new problem, an explanation task and an interleaved related skill. It is not an official IEM question paper, a difficulty forecast or a prescribed timed examination.

Write an attempt first, then compare your method with the visible answers. Record whether a miss came from interpretation, a missing concept, execution or a condition you overlooked.

Question 1: apply the topic

Integrate 3(3x+1)² dx.

Answer and checking point

With u=3x+1, the result is (3x+1)³/3+C.

Substitute a candidate result into the original statement where possible. Inspect sign, endpoint inclusion and whether a real or complex answer is requested. The simplified expression and the original expression can have different domains even when their nearby values agree.

Question 2: diagnose an approach

Explain why the following warning matters and identify the correction it calls for: Replacing the expression but leaving an unmatched differential loses a required factor.

Model checkpoint: Substitution replaces a repeated inner expression and its differential. The transformed integral must use a consistent variable. Governing rule: If u=g(x), then du=g′(x)dx; change limits for definite integrals or substitute back afterward.

Question 3: transfer to integration by parts

Integrate x cos x dx.

Answer: Take u=x and dv=cos x dx; the result is x sin x+cos x+C.

This is an interleaved related-skill task, not a claim that the two topics have identical methods. Explain which idea changes when switching from the main topic to this one.

Review the attempt

Separate correct answers reached with a justified method from lucky guesses. Rework missed steps without looking at the answer, then use a fresh problem from an appropriate primary textbook or official paper. Do not use a small worksheet result as an admission cutoff, rank or readiness guarantee.

Official sources and verification

Sources checked on 26 October 2026. A source’s date, programme and campus limit which claims it supports. Official pages may change after review.

1. UEM — IEMJEE examination information

   Historical examination pattern; verify the current cycle on the central portal.

2. NTA — explicitly labelled JEE Main 2026 syllabus

   Historical PCM syllabus reference. Lessons and exercises are independently authored, not official IEMJEE papers or verified 2027 paper predictions.

3. NCERT — official textbook catalogue

   Primary textbook reference for checking school-level concepts. This directory does not certify the independent lesson or establish IEM examination weightage.

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