# CIE Calculus (Additional Mathematics -2018)

$\def\D{\displaystyle}$

1 (CIE 2012, s, paper 21, question 4)

(i) Find $\D \frac{d}{dx}(x^2\ln x).$ 

(ii) Hence, or otherwise, find $\D \displaystyle\int x\ln x dx.$ 

2 (CIE 2012, w, paper 12, question 11either)

A curve is such that $\D y = \frac{5x^2}{1+x^2}.$

(i) Show that $\D \frac{dy}{dx}=\frac{kx}{(1+x^2)^2},$ where k is an integer to be found. 

(ii) Find the coordinates of the stationary point on the curve and determine the nature of this

stationary point. 

(iii) By using your result from part (i), find $\D \displaystyle\int \frac{x}{(1+x^2)^2} dx$  and hence evaluate $\D \displaystyle\int_{-1}^{2}\frac{x}{(1+x^2)^2}dx.$

3 (CIE 2012, w, paper 13, question 11or)

(i) Given that $\D y =\frac{3e^{2x}}{1+e^{2x}},$ show that $\D \frac{dy}{dx}=\frac{Ae^{2x}}{(1+e^{2x})^2},$  where $\D A$ is a constant to be found. 

(ii) Find the equation of the tangent to the curve $\D y = \frac{3e^{2x}}{1+e^{2x}}$  at the point where the curve

crosses the y-axis. 

(iii) Using your result from part (i), find $\D \displaystyle\int \frac{e^{2x}}{(1+e^{2x})^2}dx$  and hence evaluate $\D \displaystyle\int_{0}^{\ln 3}\frac{e^{2x}}{(1+e^{2x})^2}dx$ 

4 (CIE 2012, w, paper 22, question 7)

(i) Find $\D \frac{d}{dx} (\tan 4x).$ 

(ii) Hence find $\D \displaystyle\int (1 + \sec^2 4x) dx.$ 

(iii) Hence show that $\D \displaystyle\int_{-\frac{\pi}{16}}^{\frac{\pi}{16}} (1 + \sec^2 4x) dx = k(\pi +4),$ where $\D k$ is a constant to be found. 

5 (CIE 2013, s, paper 12, question 10)

(a) (i) Find $\D \displaystyle\int \sqrt{2x-5}dx.$ 

(ii) Hence evaluate $\D \displaystyle\int_{3}^{15}\sqrt{2x-5}dx.$ 

(b) (i) Find $\D \frac{d}{dx}(x^3\ln x).$ 

(ii) Hence find $\D \displaystyle\int x^2 \ln xdx.$ 

6 (CIE 2013, s, paper 22, question 11)

A curve has equation $\D y = 3x +\frac{1}{(x-4)^3}.$

(i) Find $\D \frac{dy}{dx}$ and $\D \frac{d^2y}{dx^2}.$ 

(ii) Show that the coordinates of the stationary points of the curve are (5, 16) and (3, 8). 

(iii) Determine the nature of each of these stationary points. 

iv) Find $\D \displaystyle\int \left(3x+\frac{1}{(x-4)^3}\right)dx.$ 

(v) Hence find the area of the region enclosed by the curve, the line $\D x = 5,$ the x-axis and the line $\D x = 6 .$ 

7 (CIE 2013, w, paper 11, question 9)

(a) Differentiate $\D 4x^3 \ln(2x +1)$  with respect to x. 

(b) (i) Given that $\D y=\frac{2x}{\sqrt{x+2}},$  show that $\D \frac{dy}{dx}=\frac{x+4}{(\sqrt{x+2})^3}.$ 

(ii) Hence find $\D \displaystyle\int \frac{5x+20}{(\sqrt{x+2})^3}dx.$ 

(iii) Hence evaluate $\D \displaystyle\int_{2}^{7}\frac{5x+20}{(\sqrt{x+2})^3}dx.$ 

8 (CIE 2014, s, paper 11, question 5)

(i) Given that $\D y= e^{x^2},$  find $\D \frac{dy}{dx}.$ 

(ii) Use your answer to part (i) to find $\D \displaystyle\int xe^{x^2} dx.$ 

(iii) Hence evaluate $\D \displaystyle\int_{0}^{2}xe^{x^2}dx.$ 

9 (CIE 2014, s, paper 23, question 10)

(i) Given that $\D y=\frac{2x}{\sqrt{x^2+21}},$  show that $\D \frac{dy}{dx}=\frac{k}{\sqrt{(x^2+21)^3}},$  where $\D k$ is a constant to be found. 

(ii) Hence find $\D \displaystyle\int \frac{6}{\sqrt{(x^2+21)^3}}dx$ and evaluate $\D \displaystyle\int_{2}^{10}\frac{6}{\sqrt{(x^2+21)^3}}dx.$ 

10 (CIE 2014, w, paper 13, question 8)

(i) Given that $\D f(x) = x \ln x^3 ,$ show that $\D f'(x) = 3(1+\ln x).$  

(ii) Hence find $\D \displaystyle\int (1+\ln x)dx.$ 

(iii) Hence find $\D \displaystyle\int_{1}^{2}\ln x dx$  in the form $\D p + \ln q,$ where $\D p$ and $\D q$ are integers. 

11 (CIE 2014, w, paper 21, question 8)

(i) Given that $\D y=\frac{x^2}{2+x^2},$ show that $\D \frac{dy}{dx}=\frac{kx}{(2+x^2)^2},$ where $\D k$ is a constant to be found. 

(ii) Hence find $\D \displaystyle\int \frac{x}{(2+x^2)^2}dx.$ 

12 (CIE $2015, \mathrm{w}$, paper 23 , question 10)

(i) Given that $\frac{\mathrm{d}}{\mathrm{d} x}\left(\mathrm{e}^{2-x^{2}}\right)=k x \mathrm{e}^{2-x^{2}}$, state the value of $k$. $$

(ii) Using your result from part (i), find $\displaystyle\int 3 x \mathrm{e}^{2-x^{2}} \mathrm{~d} x$.

(iii) Hence find the area enclosed by the curve $y=3 x \mathrm{e}^{2-x^{2}}$, the $x$ -axis and the lines $x=1$ and $x=\sqrt{2}, \quad$ 

(iv) Find the coordinates of the stationary points on the curve $y=3 x \mathrm{e}^{2-x^{2}}$. 

13 (CIE 2016 , march, paper 12, question 10)

(i) Find $\frac{\mathrm{d}}{\mathrm{d} x}\left(x(2 x-1)^{\frac{3}{2}}\right)$. 

(ii) Hence, show that $\displaystyle\int x(2 x-1)^{\frac{1}{2}} \mathrm{~d} x=\frac{(2 x-1)^{\frac{3}{2}}}{15}(p x+q)+c$, where $c$ is a constant of integration, and $p$ and $q$ are integers to be found.

(iii) Hence find $\displaystyle\int_{0.5}^{1} x(2 x-1)^{\frac{1}{2}} \mathrm{~d} x$. 

14 (CIE 2016, s, paper 11, question 5) Do not use a calculator in this question.

(i) Show that $\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{\mathrm{e}^{4 x}}{4}-x \mathrm{e}^{4 x}\right)=p x \mathrm{e}^{4 x}$, where $p$ is an integer to be found. $$

(ii) Hence find the exact value of $\displaystyle\int_{0}^{1 \mathrm{n} 2} x \mathrm{e}^{4 \mathrm{x}} \mathrm{d} x$, giving your answer in the form $a \ln 2+\frac{b}{c}$, where $a, b$ and $c$ are integers to be found. $$

15 (CIE 2016, w, paper 13, question 6)

(i) Find $\frac{\mathrm{d}}{\mathrm{d} x}\left(\ln \left(3 x^{2}-11\right)\right)$.

(ii) Hence show that $\displaystyle\int \frac{x}{3 x^{2}-11} \mathrm{~d} x=p \ln \left(3 x^{2}-11\right)+c$, where $p$ is a constant to be found, and $c$ is a constant of integration.

(iii) Given that $\displaystyle\int_{2}^{a} \frac{x}{3 x^{2}-11} \mathrm{~d} x=\ln 2$, where $a>2$, find the value of $a$. $$

16 (CIE $2016, \mathrm{w}$, paper 21 , question 8) The function $\mathrm{f}(x)$ is given by $\mathrm{f}(x)=\frac{3 x^{3}-1}{x^{3}+1}$ for $0 \leqslant x \leqslant 3$.

(i) Show that $\mathrm{f}^{\prime}(x)=\frac{k x^{2}}{\left(x^{3}+1\right)^{2}}$, where $k$ is a constant to be determined. 

(ii) Find $\displaystyle\int \frac{x^{2}}{\left(x^{3}+1\right)^{2}} \mathrm{~d} x$ and hence evaluate $\displaystyle\int_{1}^{2} \frac{x^{2}}{\left(x^{3}+1\right)^{2}} \mathrm{~d} x$.

(iii) Find $\mathrm{f}^{-1}(x)$, stating its domain.

17 (CIE 2017, march, paper 22, question 9)

(a) Find $\displaystyle\int \mathrm{e}^{2 x+1} \mathrm{~d} x$.

(b) (i) Given that $y=\frac{x}{\ln x}$, find $\frac{\mathrm{d} y}{\mathrm{~d} x}$.

(ii) Hence find $\displaystyle\int\left(\frac{1}{\ln x}-\frac{1}{(\ln x)^{2}}+\frac{1}{x^{2}}\right) \mathrm{d} x$. 

18 (CIE 2017 , s, paper 11 , question 9)

(i) Show that $5+4 \tan ^{2}\left(\frac{x}{3}\right)=4 \sec ^{2}\left(\frac{x}{3}\right)+1$. $$

(ii) Given that $\frac{\mathrm{d}}{\mathrm{d} x}\left(\tan \left(\frac{x}{3}\right)\right)=\frac{1}{3} \sec ^{2}\left(\frac{x}{3}\right)$, find $\displaystyle\int \sec ^{2}\left(\frac{x}{3}\right) \mathrm{d} x$. $$

(iii) The diagram shows part of the curve $y=5+4 \tan ^{2}\left(\frac{x}{3}\right)$. Using the results from parts (i) and (ii), find the exact area of the shaded region enclosed by the curve, the $x$ -axis and the lines $x=\frac{\pi}{2}$ and $x=\pi .$

19 (CIE 2017, s, paper 12 , question 11) The curve $y=\mathrm{f}(x)$ passes through the point $\left(\frac{1}{2}, \frac{7}{2}\right)$ and is such that $\mathrm{f}^{\prime}(x)=\mathrm{e}^{2 x-1}$.

(i) Find the equation of the curve. $$

(ii) Find the value of $x$ for which $f^{\prime \prime}(x)=4$, giving your answer in the form $a+b \ln \sqrt{2}$, where $a$ and $b$ are constants.

20 (CIE 2017, s, paper 13 , question 9 )

It is given that $\displaystyle\int_{-k}^{k}\left(15 e^{5 x}-5 e^{-5 x}\right) \mathrm{d} x=6$.

(i) Show that $\mathrm{e}^{5 k}-\mathrm{e}^{-5 k}=3$. $$

(ii) Hence, using the substitution $y=\mathrm{e}^{5 k}$, or otherwise, find the value of $k$ $$

21 (CIE 2017, s, paper 13, question 10) It is given that $y=(10 x+2) \ln (5 x+1)$.

(i) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$. 

(ii) Hence show that $\displaystyle\int \ln (5 x+1) \mathrm{d} x=\frac{(a x+b)}{5} \ln (5 x+1)-x+c$, where $a$ and $b$ are integers and $c$ is a constant of integration. 

(iii) Hence find $\displaystyle\int_{0}^{\frac{1}{5}} \ln (5 x+1) \mathrm{d} x$, giving your answer in the form $\frac{d+\ln f}{5}$, where $d$ and $f$ are integers. $$

22 (CIE 2017, s, paper 22, question 5)

(i) Show that $\frac{\mathrm{d}}{\mathrm{d} x}\left[0.4 x^{5}(0.2-\ln 5 x)\right]=k x^{4} \ln 5 x$, where $k$ is an integer to be found. 

(ii) Express $\ln 125 x^{3}$ in terms of $\ln 5 x$. $$

(iii) Hence find $\displaystyle\int\left(x^{4} \ln 125 x^{3}\right) \mathrm{d} x$. 

23 (CIE $2017, \mathrm{w}$, paper 13 , question 2) A curve is such that its gradient at the point $(x, y)$ is given by $10 \mathrm{e}^{5 x}+3$. Given that the curve passes though the point $(0,9)$, find the equation of the curve.

24 (CIE 2017, w, paper 21, question 5)

(i) Find $\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{5}{3 x+2}\right)$. 

(ii) Use your answer to part (i) to find $\displaystyle\int \frac{30}{(3 x+2)^{2}} \mathrm{~d} x$. 

(iii) Hence evaluate $\displaystyle\int_{1}^{2} \frac{30}{(3 x+2)^{2}} \mathrm{~d} x$. $$

25 (CIE 2017, w, paper 22, question 9)

(i) Find $\frac{\mathrm{d}}{\mathrm{d} x}(x \ln x)$. $$

(ii) Hence find $\displaystyle\int \ln x \mathrm{~d} x$. 

(iii) Hence, given that $k>0$, show that $\displaystyle\int_{k}^{2 k} \ln x \mathrm{~d} x=k(\ln 4 k-1)$. 

26 (CIE 2017, w, paper 23 , question 9)

(i) Show that $\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{\ln x}{x^{3}}\right)=\frac{1-3 \ln x}{x^{4}}$. 

(ii) Find the exact coordinates of the stationary point of the curve $y=\frac{\ln x}{r^{3}}$. $$

(iii) Use the result from part (i) to find $\displaystyle\int\left(\frac{\ln x}{x^{4}}\right) \mathrm{d} x$. 

27 (CIE 2018, march, paper 12 , question 10)

The diagram shows the graph of $y=(x+2)^{2}(1-3 x)$. The curve has a minimum at the point $A$, a maximum at the point $B$ and intersects the $y$ -axis and the $x$ -axis at the points $C$ and $D$ respectively.

(i) Find the $x$ -coordinate of $A$ and of $B$. $$

(ii) Write down the coordinates of $C$ and of $D$. 

(iii) Showing all your working, find the area of the shaded region. 

28 (CIE 2018, march, paper 22, question 6)

(i) Differentiate $1+\tan \left(\frac{x}{3}\right)$ with respect to $x$. 

(ii) Hence find $\displaystyle\int \sec ^{2}\left(\frac{x}{3}\right) \mathrm{d} x$. 

1. (i) $\D 2x \ln x + x$

(ii) $\D 0.5x^2 \ln x - x^2/4$

2. (i) $\D k = 10$

(ii) $\D (0,0),$ min

(iii) $\D \frac{x^2}{2(1+x^2)},$

$\D 0.15$

3. (i) $\D A = 6$

(ii) $\D 2y - 3 = 3x$

(iii) $\D \frac{e^{2x}}{2(1+e^{2x})}, 0.2$

4. (i) $\D 4 \sec^2 4x$

(ii) $\D x + \frac{1}{4}\tan 4x$

(iii) $\D k=1/8$

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

(ii) $\D 124/3$

(b)(i) $\D x^2 + 3x^2 \ln x$

(ii) $\D \frac{1}{3}(x^3 \ln x - \frac{x^3}{3})$

6. (i) $\D y'= 3-3(x-4)^{-4}$

$\D y''=12(x-4)^{-5}$

(iii) $\D x = 5$, min, $\D x = 3,$ max

(iv) $\D \frac{3x^2}{2}-\frac{(x-4)^2}{2}$

(v) $\D 135/8$

7. (a) $\D 12x^2 \ln(2x+1)+8x^3/(2x+1)$

(b)(ii) $\D \frac{10x}{\sqrt{x+2}}$

(iii) $\D 40/3$

8. $\D 2xe^{x^2},0.5e^{x^2},26.8$

9. (i) $\D k = 42$

(ii) $\D \frac{8}{55}$

10. (ii) $\D x \ln x$

(iii) $\D -1 + \ln 4$

11. (i) $\D k = 4$

(ii) $\D \frac{x^2}{4(2+x^2)}$

12. (i) $k=-2$

(ii) $(-3 / 2) e^{2-x^{2}}$

(iii) $2.58$

(iv) $x=\pm 0.707, y=\pm 9.51$

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

(ii) $p=3, q=1($ iii $) 4 / 15$

14. (i) $-4 x e^{4 x}$

(ii) $4 \ln 2-15 / 16$

15. (i) $6 x /\left(3 x^{2}-11\right)$

(ii) $p=1 / 6$, (iii) $a=5$

16. (i) $k=12(\mathrm{ii}) 7 / 54$

(iii) $f^{-1}(x)=\sqrt{\frac{x+1}{3-x}}$

$\mathrm{D}:-1 \leq x \leq 20 / 7$

17. (a). $5 e^{2 x+1}(+\mathrm{c})$

(bi) $(\ln x-1) /(\ln x)^{2}$

(bii) $x / \ln x-1 / x(+c)$

18. (ii) $3 \tan (\mathrm{x} / 3)$

(iii) $8 \sqrt{3}+\pi / 2$

19. (i) $f(x)=(1 / 2) e^{2 x-1}+3$

(ii) $x=1 / 2+\ln \sqrt{2}$

20. $k=0.239$

21. (i) $(10 x+2) \times \frac{5}{5 x+1}+10 \ln (5 x+1)$

(ii) $\frac{5 x+1}{5} \ln (5 x+1)-x$

(iii) $\frac{-1+\ln 4}{5}$

22. (ii) $3 \ln 5 x$

(iii) $-1.5\left(0.4 x^{3}(0.2-\ln 5 x)\right)$

23. $y=2 e^{5 x}+3 x+7$

24. (i) $15(3 x+2)^{-2}$

(ii) $-10 /(3 x+2)$ (iii) $3 / 4$

25. (i) $1+\ln x$

(ii) $x \ln x-x+(c)$

(iii) $k(\ln 4 k-1)$

26. (ii) $\left(e^{1 / 3}, 1 /(3 e)\right)$

(iii) $-1 /\left(9 x^{3}\right)-\ln x /\left(3 x^{3}\right)$

27. (i) $-2,-4 / 9$

(ii) $C(0,4), D(1 / 3,0)$

(iii) $0.744$

28. $(1 / 3) \sec ^{2}(x / 3)$

$3 \tan (x / 3)+c$