Reduction formula or recurrence - Integration by Parts

Edexcel F3 2020 June Q4

9 marks Challenging +1.2

4.

Show that, for $n \geqslant 2$
Hence find the functions $\mathrm { f } ( x )$ and $\mathrm { g } ( x )$ such that $$\int x ^ { 4 } \cos x \mathrm {~d} x = \mathrm { f } ( x ) \sin x + \mathrm { g } ( x ) \cos x + c$$ where $c$ is an arbitrary constant. $$I _ { n } = \int x ^ { n } \cos x \mathrm {~d} x$$
Show that, for $n \geqslant 2$ $$I _ { n } = x ^ { n } \sin x + n x ^ { n - 1 } \cos x - n ( n - 1 ) I _ { n - 2 }$$
Hence find the functions $\mathrm { f } ( x )$ and $\mathrm { g } ( x )$ such that

CAIE FP1 2018 June Q11 EITHER

Challenging +1.8

Show that $$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { x } \cos x \mathrm {~d} x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \frac { 1 } { 2 } \pi } + \mathrm { e } ^ { - \frac { 1 } { 2 } \pi } \right)$$
It is given that, for $n \geqslant 0$, $$I _ { n } = \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \cos ^ { n } x \mathrm {~d} x$$ Show that, for $n \geqslant 2$, $$4 I _ { n } = n ( n - 1 ) \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \sin ^ { 2 } x \cos ^ { n - 2 } x \mathrm {~d} x - n I _ { n }$$ and deduce the reduction formula $$\left( n ^ { 2 } + 4 \right) I _ { n } = n ( n - 1 ) I _ { n - 2 }$$
Using the result in part (i) and the reduction formula in part (ii), find the $y$-coordinate of the centroid of the region bounded by the $x$-axis and the arc of the curve $y = \mathrm { e } ^ { x } \cos x$ from $x = - \frac { 1 } { 2 } \pi$ to $x = \frac { 1 } { 2 } \pi$. Give your answer correct to 3 significant figures.

Edexcel FP2 2020 June Q7

8 marks Challenging +1.8

7. $$I _ { n } = \int \left( 4 - x ^ { 2 } \right) ^ { - n } \mathrm {~d} x \quad n > 0$$

Show that, for $n > 0$ $$I _ { n + 1 } = \frac { x } { 8 n \left( 4 - x ^ { 2 } \right) ^ { n } } + \frac { 2 n - 1 } { 8 n } I _ { n }$$
Find $I _ { 2 }$

Edexcel FP2 2021 June Q7

15 marks Challenging +1.8

In this question you must show all stages of your working.

You must not use the integration facility on your calculator. $$I _ { n } = \int t ^ { n } \sqrt { 4 + 5 t ^ { 2 } } \mathrm {~d} t \quad n \geqslant 0$$

Show that, for $n > 1$ $$I _ { n } = \frac { t ^ { n - 1 } } { 5 ( n + 2 ) } \left( 4 + 5 t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } - \frac { 4 ( n - 1 ) } { 5 ( n + 2 ) } I _ { n - 2 }$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1241b133-4161-4c04-9b50-067904cc25c2-20_385_394_829_833} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve shown in Figure 1 is defined by the parametric equations $$x = \frac { 1 } { \sqrt { 5 } } t ^ { 5 } \quad y = \frac { 1 } { 2 } t ^ { 4 } \quad 0 \leqslant t \leqslant 1$$ This curve is rotated through $2 \pi$ radians about the $x$-axis to form a hollow open shell.
Show that the external surface area of the shell is given by $$\pi \int _ { 0 } ^ { 1 } t ^ { 7 } \sqrt { 4 + 5 t ^ { 2 } } \mathrm {~d} t$$ Using the results in parts (a) and (b) and making each step of your working clear,
determine the value of the external surface area of the shell, giving your answer to 3 significant figures.

OCR Further Additional Pure 2018 March Q6

14 marks Challenging +1.8

6 In this question you must show detailed reasoning. It is given that $I _ { n } = \int _ { 0 } ^ { \sqrt { 3 } } t ^ { n } \sqrt { 1 + t ^ { 2 } } \mathrm {~d} t$ for integers $n \geqslant 0$.

Show that $I _ { 1 } = \frac { 7 } { 3 }$.
Prove that, for $n \geqslant 2 , ( n + 2 ) I _ { n } = 8 ( \sqrt { 3 } ) ^ { n - 1 } - ( n - 1 ) I _ { n - 2 }$. The curve $C$ is defined parametrically by $$x = 10 t ^ { 3 } , y = 15 t ^ { 2 } \text { for } 0 \leqslant t \leqslant \sqrt { 3 }$$ When the curve $C$ is rotated through $2 \pi$ radians about the $x$-axis, a surface of revolution is formed with surface area $A$.
Determine
- the values of the integers $k$ and $m$ such that $A = k \pi I _ { m }$,
- the exact value of $A$.

Edexcel F3 Specimen Q4

8 marks Challenging +1.2

4. $I _ { n } = \int _ { 0 } ^ { a } ( a - x ) ^ { n } \cos x \mathrm {~d} x , \quad a > 0 , \quad n \geqslant 0$

Show that, for $n \geqslant 2$, $$I _ { n } = n a ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
Hence evaluate $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left( \frac { \pi } { 2 } - x \right) ^ { 2 } \cos x d x$

Edexcel FP3 Q5

4 marks Challenging +1.3

5. $$I _ { n } = \int _ { 0 } ^ { 5 } \frac { x ^ { n } } { \sqrt { } \left( 25 - x ^ { 2 } \right) } \mathrm { d } x , \quad n \geq 0$$

Find an expression for $\int \frac { x } { \sqrt { \left( 25 - x ^ { 2 } \right) } } \mathrm { d } x , \quad 0 \leq x \leq 5$.
Using your answer to part (a), or otherwise, show that $$I _ { n } = \frac { 25 ( n - 1 ) } { n } I _ { n - 2 } , \quad n \geq 2$$
Find $I _ { 4 }$ in the form $k \pi$, where $k$ is a fraction.

Edexcel FP3 Q9

8 marks Challenging +1.8

9. $$I _ { n } = \int \left( x ^ { 2 } + 1 \right) ^ { - n } \mathrm {~d} x , \quad n > 0$$

Show that, for $n > 0$, $$I _ { n + 1 } = \frac { x \left( x ^ { 2 } + 1 \right) ^ { - n } } { 2 n } + \frac { 2 n - 1 } { 2 n } I _ { n }$$
Find $I _ { 2 }$.

Edexcel FP3 Q4

9 marks Challenging +1.3

4. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } x ^ { n } \cos x \mathrm {~d} x , \quad n \geq 0$$

Prove that $I _ { n } = \left( \frac { \pi } { 2 } \right) ^ { n } - n ( n - 1 ) I _ { n - 2 } , n \geq 2$.
Find an exact expression for $I _ { 6 }$.
[0pt] [P5 June 2002 Qn 4]

Edexcel FP3 Q15

13 marks Challenging +1.2

15. $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x \text { and } J _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x } \mathrm {~d} x , \quad n \geq 0$$

Show that, for $n \geq 1$, $$I _ { n } = \mathrm { e } - n I _ { n - 1 } .$$
Find a similar reduction formula for $J _ { n }$.
Show that $J _ { 2 } = 2 - \frac { 5 } { \mathrm { e } }$.
Show that $\int _ { 0 } ^ { 1 } x ^ { n } \cosh x \mathrm {~d} x = \frac { 1 } { 2 } \left( I _ { n } + J _ { n } \right)$.
Hence, or otherwise, evaluate $\int _ { 0 } ^ { 1 } x ^ { 2 } \cosh x \mathrm {~d} x$, giving your answer in terms of e.
[0pt] [P5 June 2003 Qn 7]

AQA Further Paper 2 2021 June Q12

12 marks Challenging +1.2

12 The integral $S _ { n }$ is defined by $$S _ { n } = \int _ { 0 } ^ { a } x ^ { n } \sinh x \mathrm {~d} x \quad ( n \geq 0 )$$ 12

Show that for $n \geq 2$ $$S _ { n } = n ( n - 1 ) S _ { n - 2 } + a ^ { n } \cosh a - n a ^ { n - 1 } \sinh a$$
12
Hence show that $\int _ { 0 } ^ { 1 } x ^ { 4 } \sinh x d x = \frac { 9 } { 2 } e + \frac { 65 } { 2 } e ^ { - 1 } - 24$