CAIE A-Level Mathematics 1.8 Integration Question Bank

CAIE A-Level Mathematics 1.8 Integration Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise reversing differentiation, determining integration constants, evaluating definite integrals and constructing exact areas between curves, lines and coordinate axes.

Exam points

  • integrate powers and transformed brackets term by term and include the constant when required
  • substitute a known point after integration to determine the curve's constant
  • choose intersection bounds and integrate upper minus lower, reporting a positive exact area

Covered syllabus points

1.8.1—Basic integration1.8.2—Constant of integration1.8.3—Definite integrals1.8.4—Area and volume

Question 2(b)

[Maximum number: 3]

Let f(x)=4sin⁡23x\mathrm{f}(x)=4 \sin ^{2} 3 x.

Find ∫f(x)dx\int \mathrm{f}(x) \mathrm{d} x.

Question 4

[Maximum number: 6]

Find the exact value of ∫01xtan⁡−1x dx\int_{0}^{1} x \tan ^{-1} x \mathrm{~d} x.

Question 4(b)

[Maximum number: 5]

The equation of a curve is such that dy dx=kx3+2x2\frac{\mathrm{d} y}{\mathrm{~d} x}=k x^{3}+\frac{2}{x^{2}}, where k is a constant. The curve passes through the point S(2,20) and the gradient of the curve at S is 652\frac{65}{2}.

The coordinates of a point T on the curve are (1, t).

Find the value of t.

Question 5(b)

[Maximum number: 5]
Figure for Question 5(b) — CAIE A-Level Mathematics

The equation of a curve is y=4x12−xy=4 x^{\frac{1}{2}}-x. The curve has a maximum point when x=a and crosses the x-axis at the point with coordinates (b, 0), where b>0. The shaded region is bounded by the curve, the line x=a and the x-axis (see diagram).

Find the exact area of the shaded region.