# Calculus (CIE )

$\def\D{\displaystyle}$
1 (CIE 2012, s, paper 21, question 4)
(i) Find $\D \frac{d}{dx}(x^2\ln x).$ 
(ii) Hence, or otherwise, find $\D \int x\ln x dx.$ 

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 $\D \int \frac{x}{(1+x^2)^2} dx$  and hence evaluate $\D \int_{-1}^{2}\frac{x}{(1+x^2)^2}dx.$

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 $\D \int \frac{e^{2x}}{(1+e^{2x})^2}dx$  and hence evaluate $\D \int_{0}^{\ln 3}\frac{e^{2x}}{(1+e^{2x})^2}dx$ 

4 (CIE 2012, w, paper 22, question 7)
(i) Find $\D \frac{d}{dx} (\tan 4x).$ 
(ii) Hence find $\D \int (1 + \sec^2 4x) dx.$ 
(iii) Hence show that $\D \int_{-\frac{\pi}{16}}^{\frac{\pi}{16}} (1 + \sec^2 4x) dx = k(\pi +4),$ where $\D k$ is a constant to be found. 

5 (CIE 2013, s, paper 12, question 10)
(a) (i) Find $\D \int \sqrt{2x-5}dx.$ 
(ii) Hence evaluate $\D \int_{3}^{15}\sqrt{2x-5}dx.$ 
(b) (i) Find $\D \frac{d}{dx}(x^3\ln x).$ 
(ii) Hence find $\D \int x^2 \ln xdx.$ 

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 $\D \int \left(3x+\frac{1}{(x-4)^3}\right)dx.$ 
(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 $\D \int \frac{5x+20}{(\sqrt{x+2})^3}dx.$ 
(iii) Hence evaluate $\D \int_{2}^{7}\frac{5x+20}{(\sqrt{x+2})^3}dx.$ 

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

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

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 $\D \int (1+\ln x)dx.$ 
(iii) Hence find $\D \int_{1}^{2}\ln x dx$  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 $\D \int \frac{x}{(2+x^2)^2}dx.$ 

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