Integration (Edexcel IGCSE Further Pure Maths): Exam Questions

Show that $\cos (A–B)–\cos (A+B)=2\sin A\sin B$

Hence express $2\sin 5x\sin 3x$ in the form $\cos mx–\cos nx$where $m$ and $n$ are integers, giving the value of $m$ and the value of $n$,

(i) Find $\int 4\sin 5\theta \sin 3\theta \mathrm{d}\theta$

(ii) Hence evaluate ${\int }_0^{\frac{\mathrm{\pi }}{6}}4\sin 5\theta \sin 3\theta \mathrm{d}\theta$, giving your answer in the form $\frac{a\sqrt{b}}{c}$ where $a$, $b$ and $c$ are integers.

Using formulae (opens in a new tab), show that

(i) $\cos 2A=2{\cos }^{2}A-1$

(3)

(ii) $\sin 2A=2\sin A\cos A$

(1)

Show that ${\cos }^{3}A=\frac{\cos 3A+3\cos A}{4}$

Hence, or otherwise, solve, giving exact values in terms of $\pi$

$8{\cos }^3\left(\frac{\theta }{2}\right)-6\cos \left(\frac{\theta }{2}\right)-1=0$ for $0⩽\theta ⩽2\mathrm{\pi }$

use algebraic integration to find the exact value of

${\int }_0^{\frac{\mathrm{\pi }}{6}}\left(4{\cos }^3\theta -\sin 2\theta \right)\mathrm{d}\theta$

Using a formula (opens in a new tab), show that

$\cos 2\theta =2{\cos }^2\theta -1$

Using a formula (opens in a new tab), show that

$\cos 2\theta =2{\cos }^2\theta -1$

Hence show that

${\int }_{\frac{\pi }{3}}^{\frac{3\pi }{4}}\left(2{\cos }^2\theta -1\right) d\theta =-\frac{a+\sqrt{b}}{c}$

where $a$, $b$ and $c$ are integers to be found.

Figure 3 shows part of the curve $C_1$ with equation $y=2{\cos }^2\theta -1$and part of the curve $C_2$ with equation $y=-\cos \theta$

Point $B$ is the intersection of $C_1$ and $C_2$ as shown in Figure 3

Point $A$$\left(\frac{3\mathrm{\pi }}{4},0\right)$ is the intersection of $C_1$ with the $\theta$-axis as shown in Figure 3

Point $E$ $\left(\frac{\pi }{2},0\right)$ is the intersection of $C_2$ with the $\theta$-axis as shown in Figure 3

The finite region $R$, shown shaded in Figure 3, is bounded by the $\theta$-axis, $C_1$ and $C_2$

Use calculus to find, in its simplest form, the exact area of $R$

Using formulae (opens in a new tab) , show that

(i) $\sin 2A=2\sin A\cos A$

(ii) $\cos 2A=2{\cos }^{2}A-1$

$f\left(\theta \right)=\frac{2\tan \theta }{1+{\tan }^2\theta }$

Show that $\mathrm{f}\left(\theta \right)=\sin 2\theta$

Solve, in radians to 3 significant figures, for $-\frac{\pi }{2}⩽x⩽\frac{\pi }{2}$, the equation

$5\tan \left(x+\frac{\pi }{6}\right)=\left[1+{\tan }^2\left(x+\frac{\pi }{6}\right)\right]\left[1-2{\cos }^2\left(x+\frac{\pi }{6}\right)\right]$

Using calculus, find the exact value of

${\int }_0^{\frac{\pi }{2}}\left(\frac{4\tan \theta }{1+{\tan }^2\theta }-\cos 5\theta +2\right) \mathrm{d}\theta$