Monday, December 17, 2018

Calculus CIE (Differentation and Integration)

$\def\D{\displaystyle}$
1 (CIE 2012, s, paper 21, question 4)
(i) Find $\D \frac{d}{dx}(x^2\ln x).$ [2]
(ii) Hence, or otherwise, find $\D \int x\ln x dx.$ [3]

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. [4]
(ii) Find the coordinates of the stationary point on the curve and determine the nature of this
stationary point. [3]
(iii) By using your result from part (i), find $\D \int \frac{x}{(1+x^2)^2} dx$  and hence evaluate $\D \int_{-1}^{2}\frac{x}{(1+x^2)^2}dx.$[4]

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. [4]
(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. [3]
(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]

4 (CIE 2012, w, paper 22, question 7)
(i) Find $\D \frac{d}{dx} (\tan 4x).$ [2]
(ii) Hence find $\D \int (1 + \sec^2 4x) dx.$ [3]
(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. [2]

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

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 $\D \frac{d^2y}{dx^2}.$ [4]
(ii) Show that the coordinates of the stationary points of the curve are (5, 16) and (3, 8). [2]
(iii) Determine the nature of each of these stationary points. [2]
iv) Find $\D \int \left(3x+\frac{1}{(x-4)^3}\right)dx.$ [2]
(v) Hence find the area of the region enclosed by the curve, the line $\D x = 5,$ the x-axis and the
line $\D x = 6 .$ [2]

7 (CIE 2013, w, paper 11, question 9)
(a) Differentiate $\D 4x^3 \ln(2x +1)$  with respect to x. [3]
(b) (i) Given that $\D y=\frac{2x}{\sqrt{x+2}},$  show that $\D \frac{dy}{dx}=\frac{x+4}{(\sqrt{x+2})^3}.$ [4]
(ii) Hence find $\D \int \frac{5x+20}{(\sqrt{x+2})^3}dx.$ [2]
(iii) Hence evaluate $\D \int_{2}^{7}\frac{5x+20}{(\sqrt{x+2})^3}dx.$ [2]

8 (CIE 2014, s, paper 11, question 5)
(i) Given that $\D y= e^{x^2},$  find $\D \frac{dy}{dx}.$ [2]
(ii) Use your answer to part (i) to find $\D \int xe^{x^2} dx.$ [2]
(iii) Hence evaluate $\D \int_{0}^{2}xe^{x^2}dx.$ [2]

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. [5]
(ii) Hence find $\D \int \frac{6}{\sqrt{(x^2+21)^3}}dx$ and evaluate $\D \int_{2}^{10}\frac{6}{\sqrt{(x^2+21)^3}}dx.$ [3]

10 (CIE 2014, w, paper 13, question 8)
(i) Given that $\D f(x) = x \ln x^3 ,$ show that $\D f'(x) = 3(1+\ln x).$  [3]
(ii) Hence find $\D \int (1+\ln x)dx.$ [2]
(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. [3]

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. [3]
(ii) Hence find $\D \int \frac{x}{(2+x^2)^2}dx.$ [2]

Answer

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

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