Differentation and Integration (CIE) (2015-2018)



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

(i) Given that $\dfrac{\mathrm{d}}{\mathrm{d} x}\left(\mathrm{e}^{2-x^{2}}\right)=k x \mathrm{e}^{2-x^{2}}$, state the value of $k$. [1]

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

(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 $x=\sqrt{2} . \quad[2]$

(iv) Find the coordinates of the stationary points on the curve $y=3 x \mathrm{e}^{2-x^{2}}$. $[4]$

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

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

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

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

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

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

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

$15(\mathrm{CIE} 2016, \mathrm{w}$, paper 13, question 6$)$

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

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

(iii) Given that $\displaystyle\int_{2}^{a} \dfrac{x}{3 x^{2}-11} \mathrm{~d} x=\ln 2$, 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 $\quad \mathrm{f}(x)=\dfrac{3 x^{3}-1}{x^{3}+1}$ for $0 \leqslant x \leqslant 3$.

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

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

(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 $y=\dfrac{x}{\ln x}$, find $\dfrac{\mathrm{d} y}{\mathrm{~d} x}$. [3]

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

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

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

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


(iii) The diagram shows part of the curve $y=5+4 \tan ^{2}\left(\dfrac{x}{3}\right)$. 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 $x=\dfrac{\pi}{2}$ and $x=\pi$.

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

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

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

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

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

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

(ii) Hence, using the substitution $y=\mathrm{e}^{5 k}$, or otherwise, find the value of $k$. $[3]$


21 (CIE 2017, s, paper 13, question 10)

It is given that $y=(10 x+2) \ln (5 x+1)$.

(i) Find $\dfrac{\mathrm{d} y}{\mathrm{~d} x}$. [4]

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

(iii) Hence find $\displaystyle\int_{0}^{\frac{1}{3}} \ln (5 x+1) \mathrm{d} x$, giving your answer in the form $\dfrac{d+\ln f}{5}$, where $d$ and $f$ are integers. [2]

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

(i) Show that $\dfrac{\mathrm{d}}{\mathrm{d} x}\left[0.4 x^{5}(0.2-\ln 5 x)\right]=k x^{4} \ln 5 x$, where $k$ is an integer to be found. [2]

(ii) Express $\ln 125 x^{3}$ in terms of $\ln 5 x$. [1]

(iii) Hence find $\displaystyle\int\left(x^{4} \ln 125 x^{3}\right) \mathrm{d} x$. [2]

23 (CIE 2017, 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. $[4]$

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

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

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

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

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

(i) Find $\dfrac{\mathrm{d}}{\mathrm{d} x}(x \ln x)$. [2]

(ii) Hence find $\displaystyle\int \ln x \mathrm{~d} x$. [2]

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

26 (CIE $2017, \mathrm{w}$, paper 23 , question 9)

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

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

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


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 $B$. [5]

(ii) Write down the coordinates of $C$ and of $D$. $[2]$

(iii) Showing all your working, find the area of the shaded region. [5]

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

(i) Differentiate $1+\tan \left(\dfrac{x}{3}\right)$ with respect to $x$. $[2]$

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

Answers

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

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

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

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

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

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

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

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