[Maximum number: 5]

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

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

[Maximum number: 5]

Show that $\int_{0}^{\pi} x^{2} \sin x \mathrm{~d} x=\pi^{2}-4$ .

(a)

Hence find the exact value of $\int_{0}^{\frac{1}{4} \pi}(\sec x+\cos x)^{2} \mathrm{~d} x$ .

[ 4 ]

[Maximum number: 5]

The diagram shows the curve $y=2+\mathrm{e}^{-2 x}$ . 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.

(a)

Find $\int 4 \cos ^{2}\left(\frac{1}{2} \theta\right) \mathrm{d} \theta$ .

[ 2 ]

(b)

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

[ 4 ]

(a)

Hence show that

\int_{\frac{1}{4} \pi}^{\frac{1}{3} \pi} 12 \sin ^{2} x \cos ^{2} x \mathrm{~d} x=\frac{\pi}{8}+\frac{3 \sqrt{ } 3}{16}

(a)

Find $\int \frac{\ln x}{x^{3}} \mathrm{~d} x$ .

[ 3 ]

(b)

Hence show that $\int_{1}^{2} \frac{\ln x}{x^{3}} \mathrm{~d} x=\frac{1}{16}(3-\ln 4)$ .

[ 2 ]

(a)

Hence show that $\int_{-\frac{1}{6} \pi}^{\frac{1}{6} \pi} \cos 3 x \cos x \mathrm{~d} x=\frac{3}{8} \sqrt{ } 3$ .

[ 3 ]

[Maximum number: 4]

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

(a)

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

[ 4 ]