Integration Using Polynomial Division

Questions that require performing polynomial division first, then integrating the resulting expression (quotient plus remainder term), often giving answers in the form a + ln(b).

8 questions · Standard +0.3

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CAIE P2 2020 June Q7

12 marks Standard +0.3

Find the quotient when $9 x ^ { 3 } - 6 x ^ { 2 } - 20 x + 1$ is divided by ( $3 x + 2$ ), and show that the remainder is 9 .
Hence find $\int _ { 1 } ^ { 6 } \frac { 9 x ^ { 3 } - 6 x ^ { 2 } - 20 x + 1 } { 3 x + 2 } \mathrm {~d} x$, giving the answer in the form $a + \ln b$ where $a$ and $b$ are integers.
Find the exact root of the equation $9 e ^ { 9 y } - 6 e ^ { 6 y } - 20 e ^ { 3 y } - 8 = 0$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE P2 2024 June Q5

8 marks Standard +0.3

5 The polynomial $\mathrm { p } ( x )$ is defined by $\mathrm { p } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } + 5 x + 4$.

Find the quotient when $\mathrm { p } ( x )$ is divided by $( 3 x + 2 )$, and show that the remainder is 6 .
Find the value of $\int _ { 0 } ^ { 2 } \frac { \mathrm { p } ( x ) } { 3 x + 2 } \mathrm {~d} x$, giving your answer in the form $\mathrm { a } + \operatorname { lnb }$ where $a$ and $b$ are integers. \includegraphics[max width=\textwidth, alt={}, center]{6ee58f43-831d-402c-9f9a-2b247b2f7ffc-10_414_693_276_687} The diagram shows the curve with equation $\mathrm { y } = \frac { \ln ( 2 \mathrm { x } + 1 ) } { \mathrm { x } + 3 }$. The curve has a maximum point $M$.

CAIE P2 2024 June Q5

8 marks Standard +0.3

5 The polynomial $\mathrm { p } ( x )$ is defined by $\mathrm { p } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } + 5 x + 4$.

Find the quotient when $\mathrm { p } ( x )$ is divided by $( 3 x + 2 )$, and show that the remainder is 6 . \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-08_2713_33_146_2012} \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-09_2723_33_138_20}
Find the value of $\int _ { 0 } ^ { 2 } \frac { \mathrm { p } ( x ) } { 3 x + 2 } \mathrm {~d} x$ ,giving your answer in the form $a + \ln b$ where $a$ and $b$ are integers.

CAIE P2 2022 November Q6

9 marks Standard +0.3

6 The polynomial $\mathrm { p } ( x )$ is defined by $$\mathrm { p } ( x ) = 12 x ^ { 3 } - 9 x ^ { 2 } + 8 x - 4$$

Find the quotient when $\mathrm { p } ( x )$ is divided by $( 4 x - 3 )$ and show that the remainder is 2 .
Hence find $\int _ { 2 } ^ { 12 } \left( \frac { \mathrm { p } ( x ) } { 4 x - 3 } - 3 x ^ { 2 } \right) \mathrm { d } x$, giving your answer in the form $a + \ln b$.

CAIE P2 2019 June Q5

8 marks Standard +0.3

Find the quotient and remainder when $2 x ^ { 3 } + x ^ { 2 } - 8 x$ is divided by ( $2 x + 1$ ).
Hence find the exact value of $\int _ { 0 } ^ { 3 } \frac { 2 x ^ { 3 } + x ^ { 2 } - 8 x } { 2 x + 1 } \mathrm {~d} x$, giving the answer in the form $\ln \left( k \mathrm { e } ^ { a } \right)$ where $k$ and $a$ are constants.

CAIE P3 2020 June Q5

8 marks Standard +0.3

Find the quotient and remainder when $2 x ^ { 3 } - x ^ { 2 } + 6 x + 3$ is divided by $x ^ { 2 } + 3$.
Using your answer to part (a), find the exact value of $\int _ { 1 } ^ { 3 } \frac { 2 x ^ { 3 } - x ^ { 2 } + 6 x + 3 } { x ^ { 2 } + 3 } \mathrm {~d} x$.

CAIE P3 2022 March Q8

8 marks Standard +0.3

Find the quotient and remainder when $8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7$ is divided by $4 x ^ { 2 } + 1$.
Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7 } { 4 x ^ { 2 } + 1 } \mathrm {~d} x$.

CAIE P3 2024 November Q9

8 marks Standard +0.3

Find the quotient and remainder when $x ^ { 4 } + 16$ is divided by $x ^ { 2 } + 4$.
Hence show that $\int _ { 2 } ^ { 2 \sqrt { 3 } } \frac { x ^ { 4 } + 16 } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 4 } { 3 } ( \pi + 4 )$.