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Home 2022 YIJC P2 Q1 It is given that $I=int{frac{x}{sqrt{4-2x}},text{d}x}$. (a) Use integration by parts to find an expression for $I$. [2] (b) Use the substitution $u=4-2x$ to find another
Home 2022 NJC P1 Q1 (i) Given that ${{I}_{n}}=int_{1}^{text{e}}{x{{left( ln x right)}^{n}},text{d}x}$ for $nin mathbb{Z}$, $nge 0$, show that ${{I}_{n}}=frac{{{text{e}}^{2}}}{2}-frac{n}{2}{{I}_{n-1}}$ for all $nin {{mathbb{Z}}^{+}}$. [2] (ii) Find the exact
Home 2022 DHS J2 MYE P1 Q2 (a) Find $int{frac{1}{left( ax-b right)left( ax+3b right)}},text{d}x$, where $a$ and $b$ are positive constants. [3] (b) Use the substitution $u=sqrt{{{x}^{3}}}$ to find the
Home 2011 MI P2 Q2 (i) Sketch the graphs of $y=frac{2}{1+{{left( x-2 right)}^{2}}}$ and $y=1$ on the same diagram, showing clearly the coordinates of the stationary point and points of
Home 2022 EJC J2 MYE P2 Q1 It is given that the origin $O$ lies on the curve $C$, which has equation $y=text{f}left( x right)$. The region ${{R}_{1}}$ is bounded
Home 2018 MI P2 Q1 In the diagram, the region $R$ is bounded by the curves $y=-{{x}^{2}}+3x-1$, $y=sqrt{x}$ and the $y$-axis. Without using a graphing calculator, find the volume of
2018 SRJC P2 Q1 $R$ is the region enclosed by the line $y=-5$ and the curves $y=-{{x}^{2}}+2x-5$ and $frac{{{x}^{2}}}{4}+frac{{{left( y+5 right)}^{2}}}{16}=1$ as shown in the diagram below. Find the volume
Home 2018 EJC P1 Q1 The shaded region $R$ bounded by the curve $x=5-{{left( y-1 right)}^{2}}$, the line $x=5-{{left( y-1 right)}^{2}}$ and the $x$-axis is rotated about the $x$-axis through
Home 2022 ASRJC J2 MYCT P1 Q12 (a) Find $int{xtext{ cose}{{text{c}}^{2}}}3xtext{ d}x$. [2] (b) The diagram shows region $R$, which is bounded by the line $x=4$, the curves $y=frac{1}{x}+2$ and
Home 2022 ACJC J2 MYE Q2 (a) If $1<a<b$, solve $int_{0}^{b}{left| a-{{x}^{2}} right|}text{ d}x$, leaving your answer in terms of $a$ and $b$. [3] (b) Without the use of a