Derive reduction formula by integration by parts

A question is this type if and only if it asks to derive a reduction formula using integration by parts, typically for integrals involving products like x^n·f(x) or trigonometric powers.

7 questions · Challenging +1.3

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CAIE Further Paper 2 2020 June Q2

6 marks Challenging +1.3

2 Let $\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 1 } ( 1 + 3 \mathrm { x } ) ^ { \mathrm { n } } \mathrm { e } ^ { - 3 \mathrm { x } } \mathrm { dx }$, where $n$ is an integer.

Show that $3 \mathrm { I } _ { \mathrm { n } } = 1 - 4 ^ { \mathrm { n } } \mathrm { e } ^ { - 3 } + 3 \mathrm { nl } _ { \mathrm { n } - 1 }$.
Find the exact value of $I _ { 2 }$.

Edexcel F3 2021 January Q6

10 marks Challenging +1.2

6. $$I _ { n } = \int \frac { x ^ { n } } { \sqrt { x ^ { 2 } + 3 } } \mathrm {~d} x \quad n \in \mathbb { N }$$

Show that $$I _ { n } = \frac { x ^ { n - 1 } } { n } \left( x ^ { 2 } + 3 \right) ^ { \frac { 1 } { 2 } } - \frac { 3 ( n - 1 ) } { n } I _ { n - 2 } \quad n \geqslant 3$$
Hence show that $$\int \frac { x ^ { 5 } } { \sqrt { x ^ { 2 } + 3 } } \mathrm {~d} x = \frac { 1 } { 5 } \left( x ^ { 2 } + 3 \right) ^ { \frac { 1 } { 2 } } \left( x ^ { 4 } + p x ^ { 2 } + q \right) + k$$ where $p$ and $q$ are integers to be determined and $k$ is an arbitrary constant.

OCR FP2 2010 June Q5

8 marks Challenging +1.2

5 It is given that, for $n \geqslant 0$, $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } } ( 1 - 2 x ) ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x$$

Prove that, for $n \geqslant 1$, $$I _ { n } = 2 n I _ { n - 1 } - 1$$
Find the exact value of $I _ { 3 }$.

CAIE FP1 2019 June Q4

8 marks Challenging +1.2

4 It is given that, for $n \geqslant 0$, $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { x ^ { 3 } } \mathrm {~d} x$$

Show that $I _ { 2 } = \frac { 1 } { 3 } ( \mathrm { e } - 1 )$.
Show that, for $n \geqslant 3$, $$3 I _ { n } = \mathrm { e } - ( n - 2 ) I _ { n - 3 }$$
Hence find the exact value of $I _ { 8 }$.

CAIE FP1 2002 November Q4

7 marks Challenging +1.2

4 It is given that, for $n \geqslant 0$, $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } e ^ { - x ^ { 2 } } d x$$

Find $I _ { 1 }$ in terms of c .
Show that $$I _ { n + 2 } = \frac { n + 1 } { 2 } I _ { n } - \frac { 1 } { 2 \mathrm { e } }$$
Find $I _ { 5 }$ in terms of $e$.

CAIE FP1 2010 November Q5

8 marks Challenging +1.2

5 Let $I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { n } \sin x \mathrm {~d} x$ for $n \geqslant 0$. Show that $$I _ { n + 2 } = 1 - ( n + 1 ) ( n + 2 ) I _ { n }$$ Hence find the value of $I _ { 6 }$, correct to 4 decimal places.

CAIE FP1 2019 November Q3

7 marks Challenging +1.8

3 The integral $I _ { n }$, where $n$ is a positive integer, is defined by $$I _ { n } = \int _ { \frac { 1 } { 2 } } ^ { 1 } x ^ { - n } \sin \pi x \mathrm {~d} x$$

Show that $$n ( n + 1 ) I _ { n + 2 } = 2 ^ { n + 1 } n + \pi - \pi ^ { 2 } I _ { n }$$
Find $I _ { 5 }$ in terms of $\pi$ and $I _ { 1 }$.