CIE Integration (Additional Mathematics -2018)

 1 (CIE 2012, s, paper 12, question 1)

(i) Find $\displaystyle\int \sqrt{7 x-5} \mathrm{~d} x$. $[3]$

(ii) Hence evaluate $\displaystyle\int_{2}^{3} \sqrt{7 x-5} \mathrm{~d} x$ $[2]$


2 (CIE 2012, w, paper 23, question 7)

(a) Find $\displaystyle\int(x+3) \sqrt{x} \mathrm{~d} x$.

(b) Find $\displaystyle\int \frac{20}{(2 x+5)^{2}} \mathrm{~d} x$ and hence evaluate $\displaystyle\int_{0}^{10} \frac{20}{(2 x+5)^{2}} \mathrm{~d} x$.


3 (CIE 2013, s, paper 11, question 5)

(i) Find $\displaystyle\int\left(1-\frac{6}{x^{2}}\right) \mathrm{d} x$.

(ii) Hence find the value of the positive constant $k$ for which $\displaystyle\int_{k}^{3 k}\left(1-\frac{6}{x^{2}}\right) \mathrm{d} x=2$.


4 (CIE 2013, w, paper 11, question 5)

(i) Find $\displaystyle\int(9+\sin 3 x) \mathrm{d} x$.

(ii) Hence show that $\displaystyle\int_{\frac{\pi}{9}}^{\pi}(9+\sin 3 x) \mathrm{d} x=a \pi+b$, where $a$ and $b$ are constants to be found.


5 (CIE 2013, w, paper 13, question 11)

(i) Given that $\displaystyle\int_{0}^{k}\left(2 \mathrm{e}^{2 x}-\frac{5}{2} \mathrm{e}^{-2 x}\right) \mathrm{d} x=\frac{3}{4}$, where $k$ is a constant, show that

$$4 e^{4 k}-12 \mathrm{e}^{2 k}+5=0$$

(ii) Using a substitution of $y=\mathrm{e}^{2 k}$, or otherwise, find the possible values of $k$.


6 (CIE 2014, s, paper 11 , question 7) A curve is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=4 x+\frac{1}{(x+1)^{2}}$ for $x>0$. The curve passes through the point $\left(\frac{1}{2}, \frac{5}{6}\right)$.

(i) Find the equation of the curve. $[4]$

(ii) Find the equation of the normal to the curve at the point where $x=1$. [4]


7 (CIE 2014, s, paper 22, question 9) A curve is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=(2 x+1)^{\frac{1}{2}}$. The curve passes through the point $(4,10)$.

(i) Find the equation of the curve. [4]

(ii) Find $\displaystyle\int y \mathrm{~d} x$ and hence evaluate $\displaystyle\int_{0}^{1.5} y \mathrm{~d} x$. [5]


8 (CIE $2014, \mathrm{w}$, paper 11, question 3$)$ A curve is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2}{\sqrt{x+3}}$ for $x>-3$, The curve passes through the point $(6,10)$.

(i) Find the equation of the curve. [4]

(ii) Find the $x$ -coordinate of the point on the curve where $y=6$. [1]


9 (CIE 2015, s, paper 12 , question 8)

(i) Find $\displaystyle\int\left(10 \mathrm{e}^{2 x}+\mathrm{c}^{-2 x}\right) \mathrm{d} x$. $[2]$

(ii) Hence find $\displaystyle\int_{-k}^{k}\left(10 \mathrm{e}^{2 x}+\mathrm{e}^{-2 n}\right) \mathrm{d} x$ in terms of the constant $k$. $[2]$

(iii) Given that $\displaystyle\int_{-k}^{k}\left(10 \mathrm{e}^{2 x}+\mathrm{e}^{-2 \pi}\right) \mathrm{d} x=-60$, show that $11 \mathrm{e}^{2 k}-11 \mathrm{e}^{-2 k}+120=0$. [2]

(iv) Using a substitution of $y=\mathrm{c}^{2 k}$ or otherwise, find the value of $k$ in the form $a \ln b$, where $a$ and $b$ are constants.


10 (CIE 2015,8 , paper 21 , question 8)

(a) (i) Find $\displaystyle\int \mathrm{e}^{4 x+3} \mathrm{~d} x$. [2]

(ii) Hence evaluate $\displaystyle\int_{25}^{3} \mathrm{e}^{4 x+3} \mathrm{~d} x$. [2]

(b) (i) Find $\displaystyle\int \cos \left(\frac{x}{3}\right) \mathrm{d} x$. $[2]$

(ii) Hence evaluate $\displaystyle\int_{0}^{\frac{\pi}{6}} \cos \left(\frac{x}{3}\right) \mathrm{d} x$.

(c) Find $\displaystyle\int\left(x^{-1}+x\right)^{2} d x$. [4]



11 (CIE 2015, w, paper 11, question 2)

A curve, showing the relationship between two variables $x$ and $y$, passes through the point $P(-1,3)$ The curve has a gradient of 2 at $P$. Given that $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=-5$, find the equation of the curve.


12 (CIE 2015, w, paper 13, question 7) A curve, showing the relationship between two variables $x$ and $y$, is such that $\frac{d^{2} y}{d x^{2}}=6 \cos 3 x$, Given that the curve has a gradient of $4 \sqrt{3}$ at the point $\left(\frac{\pi}{9},-\frac{1}{3}\right)$, find the equation of the curve.


13 (CIE 2016, march, paper 22, question 3)

Find the equation of the curve which passes through the point $(1,7)$ and for which $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{9 x^{4}-3}{x^{2}}$, [4]


14 (CIE 2016, s, paper 22, question 9)

(a) Find $\displaystyle\int \frac{x^{3}+x^{2}+1}{x^{2}} \mathrm{~d} x$. $[3]$

(b) (i) Find $\displaystyle\int \sin (5 x+\pi) \mathrm{d} x$. $[2]$

(ii) Hence evaluate $\displaystyle\int_{-\frac{r}{5}}^{0} \sin (5 x+\pi) \mathrm{d} x$. $[2]$


15 (CIE 2017, s, paper 12, question 6)

(i) Show that $\frac{\operatorname{cosec} \theta}{\cot \theta+\tan \theta}=\cos \theta$. [4]

It is given that $\displaystyle\int_{0}^{a} \frac{\operatorname{cosec} 2 \theta}{\cot 2 \theta+\tan 2 \theta} \mathrm{d} \theta=\frac{\sqrt{3}}{4}$, where $0<a<\frac{\pi}{4}$.

(ii) Using your answer to part (i) find the value of $a$, giving your answer in terms of $\pi$. [4]


16 (CIE 2017, w, paper 13, question 5)

(i) Find $\displaystyle\int(7 x-10)^{-\frac{3}{3}} \mathrm{~d} x$. [2]

(ii) Given that $\displaystyle\int_{6}^{a}(7 x-10)^{-\frac{1}{5}} \mathrm{~d} x=\frac{25}{14}$, find the exact value of $a$.


17 (CIE 2018, s, paper 11, question 12) A curve is such that $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=(2 x-5)^{-\frac{1}{2}}$. Given that the curve has a gradient of 6 at the point $\left(\frac{9}{2}, \frac{2}{3}\right)$. find the equation of the curve.


18 (CIE 2018, s, paper 12 , question 11 ) 


The diagram shows the graph of the curve $y=\frac{\mathrm{e}^{4 \mathrm{r}}+3}{8}$. The curve meets the $y$ -axis at the point $A$. The normal to the curve at $A$ meets the $x$ -axis at the point $B$. Find the area of the shaded region enclosed by the curve, the line $A B$ and the line through $B$ parallel to the $y$ -axis. Give your answer in the form $\frac{\mathrm{e}}{a}$, where $a$ is a constant. You must show all your working. $[10]$


19 (CIE 2018, s, paper 22, question 11)

(a) Find $\displaystyle\int \sqrt[3]{2 x-1} \mathrm{~d} x$. $[2]$

(b) (i) Find $\displaystyle\int \sin 4 x \mathrm{~d} x$. [2]

(ii) Hence evaluate $\displaystyle\int_{\frac{\pi}{8}}^{\frac{\pi}{4}} \sin 4 x \mathrm{~d} x$. $[2]$

(c) Show that $\displaystyle\int_{0}^{\ln 8} \mathrm{e}^{\frac{x}{3}} \mathrm{~d} x=3$. [5]


Answers


1. (i) $\frac{2}{21}(7 x-5)^{3 / 2}$

(ii) $74 / 21$

2. (a) $0.4 x^{5 / 2}+2 x^{3 / 2}$

(b) $-10 /(2x+5), 1.6$

3. (i) $x+6 / x$ (ii) $k=2$

4. (i) $9 x-(1 / 3) \cos 3 x$

(ii) $8 \pi+0.5$

5. (ii) $k=0.458,-0.347$

6. (i) $y=2 x^{2}-\frac{1}{x+1}+1$

(ii) $8 x+34 y-93=0$

7. (i) $(2 x+1)^{3 / 2} / 3+1$

(ii) $107 / 30$

8. (i) $y=4(x+3)^{1 / 2}-2$

(ii) $x=1$

9. (i) $5 e^{2 x}-.5 e^{-2 x}+(c)$

(iv) $-\ln \sqrt{11}$

10. (ai) $(1 / 4) e^{4 x+3}$

(ii) 707000

(bi) $3 \sin (x / 3)$

(ii) $0.521$

(c) $-x^{-1}+2 x+x^{3} / 3$

11. $y=5 / 2-5 x^{2} / 2-3 x$

12. $y=-\frac{2}{3} \cos 3 x+3 \sqrt{3} x-\frac{\sqrt{3}}{3} \pi$

13. $y=3 x^{3}+3 x^{-1}+1$

14. (a) $x^{2} / 2+-1 / x$

(bi) $-\cos (5 x+\pi) / 5$ (ii) $0.4$

15. $\pi / 6$

16. (i) $(5 / 14)(7 x-10)^{2 / 5}$

(ii) $253 / 7$

17. $y=\frac{1}{3}(2 x-5)^{\frac{3}{2}}+4 x-20$

18. $\frac{e}{32}$

19. $($ a $)(3 / 8)(2 x-1)^{4 / 3}$

(bi) $(-1 / 4) \cos 4 x[+c]$

(bii) $1 / 4$

(c)3


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