Free · Year 5 of high school · 50 minutes · 15 questions · no penalty

Practice maths class test — Year 5: integrals.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Note on school types. At the liceo scientifico integrals come between March and April of Year 5 and also count for the second paper; at the classical and human-sciences high schools the programme usually stops at immediate integrals and areas.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

Programme covered by the test

Immediate and almost-immediate integrals
Integration by substitution, including in definite integrals
Integration by parts and cyclic cases
Integrals of rational functions in the three cases of the discriminant
The fundamental theorem of calculus and the mean value
Areas with functions that change sign and areas between two curves
Volumes of solids of revolution about x and about y

Two worked examples from the test

Question 1 · Integrali indefiniti e definiti, aree e volumi · Integrali immediati e quasi-immediati

L'integrale indefinito

$\int 3 x^{2} d x$

è uguale a

A $x^{3}$
B $x^{3} + c$ correct answer
C $3 x^{3} + c$
D $6 x + c$

Solution

Applico la regola della potenza: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} .$
Quindi $\int 3 x^{2} d x = 3 \cdot \frac{x ^{3}}{3} = x^{3} .$
L'integrale indefinito però non è una singola funzione: è l'insieme di tutte le primitive, e due primitive differiscono per una costante.
Va perciò sempre aggiunta la costante di integrazione: $x^{3} + c$ . Senza $+ c$ la risposta è incompleta.
$3 x^{3} + c$ nasce dal non dividere per il nuovo esponente; $6 x + c$ è la derivata di $3 x^{2}$ , non una sua primitiva.

Question 2 · Integrali indefiniti e definiti, aree e volumi · Integrali immediati e quasi-immediati

Calcola, per $x > 0$ :

$\int (\frac{2}{x} + 5 x) d x$

A $2 ln x + \frac{15}{2} x^{3} + c$
B $- \frac{2}{x ^{2}} + \frac{10}{3} x^{3} + c$
C $2 ln x + \frac{5}{2} x + c$
D $2 ln x + \frac{10}{3} x^{3} + c$ correct answer

Solution

Per linearità spezzo: $\int \frac{2}{x} d x + \int 5 x d x .$
Primo addendo: $\int \frac{1}{x} d x = ln x$ (qui $x > 0$ ), quindi resta $2 ln x$ .
Secondo addendo: riscrivo $x = x^{1/2}$ e uso la regola della potenza: $\int x^{1/2} d x = \frac{x ^{3/2}}{3/2} = \frac{2}{3} x^{3} .$
Moltiplico per $5$ : $\frac{10}{3} x^{3}$ .
Risultato: $2 ln x + \frac{10}{3} x^{3} + c .$
Il coefficiente $\frac{15}{2}$ si ottiene moltiplicando per $\frac{3}{2}$ invece di dividere per $\frac{3}{2}$ .

In the online practice test you will find 13 more questions like these; the correct answer to every question arrives by email.

Prefer paper? Download the test as a PDF (15 questions, 5 pages · 97 KB, ready to print). You receive the solutions by email after taking it online.

Download the PDF

What a maths test on integrals looks like

Immediate and almost-immediate integrals, a substitution, an integration by parts (sometimes cyclic), a rational function, the calculation of an area, also between two curves, and at least one volume of revolution.

The mistake that costs the grade

In an area, integrating the function across a point where it changes sign without splitting the interval: the integral gives zero while the area is not. In a substitution inside a definite integral, forgetting to change the limits of integration too.

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.

Is the score not enough?

See which topics lost you the most marks, then read the guide to integrals explained well — or we look at your results together: the diagnostic is free.

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