Senior Secondary (HKDSE) · Mathematics M2 (Algebra and Calculus)

Definite integration: Practice Questions

5 multiple-choice questions marked as you go, and 5 written questions with worked solutions. All on Definite integration.

10 questions23 marksFree, no account

Question 1

1 mark

Evaluate the definite integral: $$ \int_1^2 \left(x^2 + \frac{1}{x}\right) dx $$

Question 2

1 mark

Evaluate the definite integral:
$$\int_0^1 x \sqrt{1-x^2} \, dx$$

Question 3

1 mark

Evaluate the definite integral:
\(\int_0^1 x \tan^{-1} x dx\)

Question 4

1 mark

Evaluate the definite integral:
\(\int_{-\pi/4}^{\pi/4} (x \cos x + \cos^2 x) dx\)

Question 5

1 mark

Evaluate the definite integral:
\( \int_0^{\pi} \frac{x \sin x}{3 + \sin^2 x} dx \)

Question 6

2 marks

Calculate the value of \(\int_{1}^{2} e^x dx\).

Write your answer out first, then check it against the worked solution.

Question 7

3 marks

Find the exact area of the region bounded by the curve $$y = \frac{1}{(2x+1)^2}$$, the x-axis, and the lines $$x=0$$ and $$x=1$$.

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Question 8

6 marks

Evaluate the definite integral: $$\int_{-2}^{2} |x^2 - 1| \, dx$$

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Question 9

3 marks

Evaluate the definite integral \( \int_{0}^{\frac{\pi}{4}} (\sec^2 x + e^{2x}) \, dx \). Give your answer in terms of \( e \) and \( \pi \).

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Question 10

4 marks

Consider the integral \( J = \int_{0}^{1} x(1-x)^5 \, dx \).
(a) By using the substitution \( u = 1-x \), or otherwise, show that \( J = \int_{0}^{1} (u^5 - u^6) \, du \).
(b) Hence, find the exact value of \( J \).

Write your answer out first, then check it against the worked solution.

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