CAIE A-Level Mathematics 1.8 Integration Question Bank

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 integration 1.8.2—Constant of integration 1.8.3—Definite integrals 1.8.4—Area and volume

[Maximum number: 5]

The function f is defined by $\mathrm{f}(x)=\frac{2}{(x+2)^{2}}$ for x>-2.

(a)

Find $\int_{1}^{\infty} \mathrm{f}(x) \mathrm{d} x$ .

[ 3 ]

$\frac{-2}{x+2}$
B1
Integrate \mathrm{f}(x). Accept $-2(x+2)^{-1}$ . Can be unsimplified.
$0-\left(-\frac{2}{3}\right)=\frac{2}{3}$
M1 A1
Apply limit(s) to an integrated expansion. CAO for A1

(b)

The equation of a curve is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\mathrm{f}(x)$ . It is given that the point (-1,-1) lies on the curve.

Find the equation of the curve.

[ 2 ]

-1=-2+c
M1
Substitute x=-1, y=-1 into their integrated expression ( c
present)
$y=\frac{-2}{x+2}+1$
A1
Accept $y=-2(x+2)^{-1}+1 .-2$ must be resolved.

[Maximum number: 4]

Figure for Question 1.8 Integration question 2 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=9\left(x^{-\frac{1}{2}}-4 x^{-\frac{3}{2}}\right)$ . The curve crosses the x-axis at the point A.

Find the area of the region bounded by the curve, the x-axis and the line x=9.

$\int 9\left(x^{-\frac{1}{2}}-4x^{-\frac{3}{2}}\right)\,dx =9\left(\frac{x^{\frac{1}{2}}}{\frac{1}{2}}-\frac{4x^{-\frac{1}{2}}}{-\frac{1}{2}}\right)$
B2,1,0: B2 for all 3 terms correct: 9, $\frac{x^{1/2}}{1/2}$ , and $\frac{-4x^{-1/2}}{-1/2}$ ; B1 for 2 of the 3 terms correct.
$9\left[\left(6+\frac{8}{3}\right)-(4+4)\right]$
M1 Apply limits $4\rightarrow 9$ to an integrated expression with no consideration of other areas.
A1 6. Use of $\pi$ scores A0.

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Covered syllabus points

1.8 Integration question 1

Question (a)

Question (b)

1.8 Integration question 2