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A-Level CAIE Mathematics A23.5 IntegrationQuestion Bank

3.5 Integration

Question 2

[Maximum number: 5]

Find the exact value of ∫01(2−x)e−2x dx\int_{0}^{1}(2-x) \mathrm{e}^{-2 x} \mathrm{~d} x.

Question 2

[Maximum number: 5]

Show that ∫0πx2sin⁡x dx=π2−4\int_{0}^{\pi} x^{2} \sin x \mathrm{~d} x=\pi^{2}-4.

Question 2

Use the substitution u=3 x+1 to find ∫3x3x+1 dx\int \frac{3 x}{3 x+1} \mathrm{~d} x.

Question 3

Question 3(a)

(a)

Find ∫4cos⁡2(12θ)dθ\int 4 \cos ^{2}\left(\frac{1}{2} \theta\right) \mathrm{d} \theta.

[ 2 ]

Question 3(b)

(b)

Find the exact value of ∫−1612x+3 dx\int_{-1}^{6} \frac{1}{2 x+3} \mathrm{~d} x.

[ 4 ]

Question 3

[Maximum number: 5]
Question image

The diagram shows the curve y=2+e−2xy=2+\mathrm{e}^{-2 x}. The curve crosses the y-axis at the point A, and the point B on the curve has x-coordinate 1 . The shaded region is bounded by the curve and the line segment A B.

Find the exact area of the shaded region.

Question 3

[Maximum number: 5]

Find the exact value of ∫14ln⁡xx dx\int_{1}^{4} \frac{\ln x}{\sqrt{ } x} \mathrm{~d} x.

Question 3

Question 3(ii)

(a)

Hence show that ∫−16π16πcos⁡3xcos⁡x dx=383\int_{-\frac{1}{6} \pi}^{\frac{1}{6} \pi} \cos 3 x \cos x \mathrm{~d} x=\frac{3}{8} \sqrt{ } 3.

[ 3 ]

Question 3

[Maximum number: 4]

It is given that x=ln⁡(1−y)−ln⁡yx=\ln (1-y)-\ln y, where 0<y<1.

Question 3(ii)

(a)

Hence show that ∫01y dx=ln⁡(2ee+1)\int_{0}^{1} y \mathrm{~d} x=\ln \left(\frac{2 \mathrm{e}}{\mathrm{e}+1}\right).

[ 4 ]

Question 3

Show that ∫01(1−x)e−12x dx=4e−12−2\int_{0}^{1}(1-x) \mathrm{e}^{-\frac{1}{2} x} \mathrm{~d} x=4 \mathrm{e}^{-\frac{1}{2}}-2.

Question 3

[Maximum number: 5]

Find the exact value of ∫012πx2sin⁡2x dx\int_{0}^{\frac{1}{2} \pi} x^{2} \sin 2 x \mathrm{~d} x.

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