Integration (Edexcel IGCSE Further Maths): Exam Questions

Figure 3 shows a sketch of the curve with equation $y=f(x)$, which passes through the points with coordinates $(-1,0),(b,0)$ and $(a,0)$ where $0<b<a.$

Given that $f'(x)=6x^{2}--26x+12$ find,

(i) the value of $a$,

(ii) the value of $b$.

Show that $\mathrm{cos}(A--B)--\mathrm{cos}(A+B)=2\mathrm{sin}A\mathrm{sin}B$

Hence express $2\mathrm{sin}5x\mathrm{sin}3x$ in the form $\mathrm{cos}mx--\mathrm{cos}nx$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 integral _ 0 ^ pi / 6 end ^ 4 sin 5 theta sin 3 theta d theta, giving your answer in the form $\frac{a\sqrt{b}}{c}$ where $a$, $b$ and $c$ are integers.

Using formulae , show that

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

(ii) $\mathrm{sin}2A=2\mathrm{sin}A\mathrm{cos}A$

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

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

$8 \text{cos} ^{3} ( \frac{ \theta }{2} ) - 6 \text{cos} ( \frac{ \theta }{2} ) - 1 = 0$ for 0 less-than or slanted equal to theta less-than or slanted equal to 2 pi

use algebraic integration to find the exact value of

integral _ 0 ^ pi / 6 end ^ ( 4 cos ^3 theta - sin 2 theta ) d theta

Using a formula, show that

$\mathrm{cos}2θ=2\mathrm{cos}^{2}θ-1$

Using a formula, show that

$\mathrm{cos}2θ=2\mathrm{cos}^{2}θ-1$

Hence show that

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

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

Figure 3 shows part of the curve $C_{1}$ with equation $y=2\mathrm{cos}^{2}θ-1$and part of the curve $C_{2}$ with equation $y=-\mathrm{cos}θ$

Point $B$ is the intersection of $C_{1}$ and $C_{2}$ as shown in Figure 3

Point $A\quad ( \frac{3 \mathrm{\pi } }{4} , 0 )$ is the intersection of $C_{1}$ with the $θ$-axis as shown in Figure 3

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

The finite region $R$, shown shaded in Figure 3, is bounded by the $θ$-axis, $C_{1}$ and $C_{2}$

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

Using formulae , show that

(i) $\mathrm{sin}2A=2\mathrm{sin}A\mathrm{cos}A$

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

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

Show that $f(θ)=\mathrm{sin}2θ$

Solve, in radians to 3 significant figures, for negative pi over 2 less-than or slanted equal to x less-than or slanted equal to pi over 2, the equation

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

Using calculus, find the exact value of

$\int_{0}^{ \frac{\pi }{2} } ( \frac{4 \tan \\theta }{1 + \tan ^{2} \\theta } - \cos 5 \\theta + 2 )   \mathrm{d} \\theta$