STEP integration: guide and practice problems

STEP integration is about choosing the right substitution or spotting the symmetry, then carrying out a long calculation without slips.

Where it comes from: Integration in A Level Pure; improper integrals, reduction formulae and volumes in Further Pure.

Key ideas

Reduction formulae: integrate by parts once, and the integral reappears with a smaller index.
Swap the limits with x -> a - x (or x -> pi/2 - x): adding the two forms often gives something trivial.
Bound an integral by bounding the integrand; then squeeze to find a limit.
Improper integrals: integrate to a finite upper limit R, then let R tend to infinity.

Common traps

Changing the variable but not the limits.
Writing ∞ into an expression instead of taking a limit.
Losing a factor of π in volumes of revolution.

Practice problems

Original problems in the style of STEP. Try each one for at least 20 minutes before you open a hint; write a full solution, then compare.

Problem 1: A reduction formula and a limit · STEP 2 style

For integers \(n\ge0\) let \(I_n=\displaystyle\int_0^1x^ne^x\,dx\).

Show that \(I_n=e-nI_{n-1}\) for \(n\ge1\), and find \(I_3\) in terms of \(e\).
Show that \(0
Deduce the value of \(\displaystyle\lim_{n\to\infty}nI_n\).

Hint 1

Integrate by parts with \(u=x^n\), \(dv=e^xdx\).

Hint 2

For (iii), rearrange the reduction formula: \(nI_{n-1}=e-I_n\). What happens to \(I_n\)?

Full solution

(i) \(I_n=\left[x^ne^x\right]_0^1-n\int_0^1x^{n-1}e^xdx=e-nI_{n-1}\). \(I_0=e-1\), \(I_1=e-(e-1)=1\), \(I_2=e-2\), \(I_3=e-3(e-2)=6-2e\).

(ii) On \((0,1)\), \(0

(iii) From (ii), \(I_n\to0\). From the reduction formula, \(nI_{n-1}=e-I_n\to e\). So \((n-1)I_{n-1}=nI_{n-1}-I_{n-1}\to e\), that is \(\lim nI_n=e\).

Results: \(I_3=6-2e\); \(\lim nI_n=e\).

2 more integration problems

Symmetry on a quarter-turn; Area and volume to infinity. Each with two hints and a full solution.

Included with A Level plans, and with IB, IGCSE or CBSE plans.

Next steps

Related topics: Sequences, series and induction · Functions and curve sketching. Stuck? Read the problem-solving strategies. Ready for the real thing? Official STEP past papers.

STEP is run by OCR. A Level Math Revision is independent: it is not affiliated with or endorsed by OCR, the University of Cambridge or any university. Every problem here is our own, written in the style of STEP; for real papers use OCR's official past papers.