Techniques of Integration

2017 IJC P1 Q2 (i) Find $int{n{{cos }^{-1}}left( nx right)}text{ d}x$, where $n$ is a positive constant. [3] (ii) Hence find the exact value of $int_{0}^{frac{1}{2n}}{n{{cos }^{-1}}left( nx right)text{ d}x}$.

2022 NJC Promo Q6 (i) Use the substitution $x=mtan t$ to find $int{frac{1}{sqrt{{{m}^{2}}+{{x}^{2}}}}text{d}x}$, where $m$ is a positive constant and $0le t<frac{pi }{2}$. [4] (ii) Find $int{frac{x}{sqrt{{{m}^{2}}+{{x}^{2}}}}text{d}x}$. [2] (iii) Hence,

2019 JPJC P1 Q9 Find (a) $int{frac{{{text{e}}^{frac{1}{x}}}}{{{x}^{2}}},text{d}x}$, [2] (b) $int{cos kxcos left( k+2 right)x,text{d}x}$, where $k$ is a positive constant, [2] (c) $int{x{{tan }^{-1}}left( 3x right),,text{d}x}$. [6] Suggested Video Solutions

2022 VJC Promo Q10 [Modified] (a) (i) Show that $9x$ can be expressed as $Aleft( 12-18x right)+B$, where $A$ and $B$ are constants to be determined. [1] (ii) Without the

Home Techniques of Integration Practice 4 Q19 By considering $x=4cos theta $ where $0le theta le pi $, find $int{sqrt{16-{{x}^{2}}},text{d}x}$. Suggested Video and handwritten Solutions Written by Did You Enjoy This

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