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IGCSE Additional Math14.13. Evaluate definite integrals and areasTopic Practice

14.13. Evaluate definite integrals and areas

CAIE IGCSE Additional Math 14.13. Evaluate definite integrals and areas question practice helps you revise this syllabus point with the course map in view. Use this page to focus on one topic, check the style of questions available, and connect each attempt back to the knowledge area it is testing.

EduNinja keeps Additional Math practice aligned to CAIE, so you can move from topic review into exam-style question bank work without losing the syllabus structure. Start with a small set, mark the weak steps, then return to nearby topic links when a definition, graph, calculation, or explanation needs repair.

Question 2(b)

[Maximum number: 2]

TRIGONOMETRY

Identities

sin⁡2A+cos⁡2A=1sec⁡2A=1+tan⁡2Acosec⁡2A=1+cot⁡2A\begin{gathered} \sin ^{2} A+\cos ^{2} A=1 \\ \sec ^{2} A=1+\tan ^{2} A \\ \operatorname{cosec}^{2} A=1+\cot ^{2} A \end{gathered}

Formulae for △ABC\triangle A B C

asin⁡A=bsin⁡B=csin⁡Ca2=b2+c2−2bccos⁡AΔ=12bcsin⁡A\begin{gathered} \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \\ a^{2}=b^{2}+c^{2}-2 b c \cos A \\ \Delta=\frac{1}{2} b c \sin A \end{gathered}

Hence find ∫0π4tan⁡2x dx\int_{0}^{\frac{\pi}{4}} \tan ^{2} x \mathrm{~d} x. Give your answer in exact form.

Question 3(b)

[Maximum number: 3]

Hence find the exact value of ∫13(4x+5−12x+3)dx\int_{1}^{3}\left(4 x+5-\frac{1}{2 x+3}\right) \mathrm{d} x, simplifying your answer.

Question 4

Find ∫02(1+e2x)2 dx\int_{0}^{2}\left(1+\mathrm{e}^{2 x}\right)^{2} \mathrm{~d} x, giving your answer in exact form.

Question 6

[Maximum number: 6]

Find the exact value of ∫23(x+2)2xdx\int_{2}^{3} \frac{(x+2)^{2}}{x} d x.
[6]

Question 6

[Maximum number: 4]

Find the exact area of the region enclosed by the curve y=e2−4xy=\mathrm{e}^{2-4 x}, the x-axis, the line x=-0.25 and the line x=0.5.

Question 9(d)

[Maximum number: 2]

The equation of a curve is y=kxe−2x\mathrm{y}=\mathrm{kxe}^{-2 \mathrm{x}}, where k is a constant.

Find the exact value of ∫014xe−2x dx\int_{0}^{1} 4 x \mathrm{e}^{-2 x} \mathrm{~d} x.

Question 6

[Maximum number: 4]

Find ∫24(22x−3−3(3x−5)2)dx\int_{2}^{4}\left(\frac{2}{2 x-3}-\frac{3}{(3 x-5)^{2}}\right) \mathrm{d} x, giving your answer in exact form.

Question 7(b)

[Maximum number: 4]

Hence find ∫123x2ln⁡x dx\int_{1}^{2} 3 x^{2} \ln x \mathrm{~d} x, giving your answer in the form ln⁡a+b\ln a+b, where a is an integer and b is a rational number.

Question 8(d)

[Maximum number: 2]

The equation of a curve is y=xsin⁡xy=x \sin x.

Evaluate ∫0π4xcos⁡x dx\int_{0}^{\frac{\pi}{4}} x \cos x \mathrm{~d} x, giving your answer correct to 2 significant figures.

Question 8(b)

[Maximum number: 4]

Show that ∫1e((1+1x)2−1)dx=3e−1e\int_{1}^{\mathrm{e}}\left(\left(1+\frac{1}{\mathrm{x}}\right)^{2}-1\right) \mathrm{dx}=\frac{3 \mathrm{e}-1}{\mathrm{e}}.

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