Trigonometric power reduction - Reduction Formulae

Edexcel F3 2023 January Q8

11 marks Challenging +1.2

8. $$I _ { n } = \int \cos ^ { n } x \mathrm {~d} x \quad n \geqslant 0$$

Prove that for $n \geqslant 2$ $$I _ { n } = \frac { 1 } { n } \cos ^ { n - 1 } x \sin x + \frac { n - 1 } { n } I _ { n - 2 }$$
Show that for positive even integers $n$ $$\int _ { 0 } ^ { \overline { 2 } } \cos ^ { n } x d x = \frac { ( n - 1 ) ( n - 3 ) \ldots 5 \times 3 \times 1 } { n ( n - 2 ) ( n - 4 ) \ldots 6 \times 4 \times 2 } \times \overline { 2 }$$
Hence determine the exact value of $$\int _ { 0 } ^ { \overline { 2 } } \cos ^ { 6 } x \sin ^ { 2 } x d x$$
GUV SIHI NI JIVM ION OC VJYV SIHI NI JIIIM ION OC VJ4V SIHIANI JIIIM ION OO

Edexcel F3 2014 June Q5

11 marks Challenging +1.8

Given that

$$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } \cos ^ { n } \theta \mathrm {~d} \theta , \quad n \geqslant 0$$

prove that, for $n \geqslant 2$, $$n I _ { n } = \left( \frac { 1 } { \sqrt { 2 } } \right) ^ { n } + ( n - 1 ) I _ { n - 2 }$$
Hence find the exact value of $I _ { 5 }$, showing each step of your working.

Edexcel F3 2017 June Q5

9 marks Challenging +1.8

5. $$I _ { n } = \int \operatorname { cosec } ^ { n } x \mathrm {~d} x , \quad 0 < x < \frac { \pi } { 2 } , \quad n \geqslant 0$$

Show that, for $n \geqslant 2$ $$I _ { n } = \frac { n - 2 } { n - 1 } I _ { n - 2 } - \frac { 1 } { n - 1 } \cot x \operatorname { cosec } ^ { n - 2 } x$$
Hence, or otherwise, find $$\int \operatorname { cosec } ^ { 4 } x \mathrm {~d} x$$ giving your answer in terms of $\cot x$.

Edexcel FP3 Q5

10 marks Challenging +1.8

5. $$\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { \mathrm { n } } x \mathrm { dx } , \mathrm { n } \geqslant 0$$

Show that $I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$, for $n \geqslant 2$
Using the result in part (a), find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \sin ^ { 5 } x \cos x d x$$

Edexcel FP3 2015 June Q7

11 marks Challenging +1.2

7. $$I _ { n } = \int \sin ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$

Prove that for $n \geqslant 2$ $$I _ { n } = \frac { 1 } { n } \left( - \sin ^ { n - 1 } x \cos x + ( n - 1 ) I _ { n - 2 } \right)$$ Given that $n$ is an odd number, $n \geqslant 3$
show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x = \frac { ( n - 1 ) ( n - 3 ) \ldots 6.4 .2 } { n ( n - 2 ) ( n - 4 ) \ldots 7.5 .3 }$$
Hence find $\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 5 } x \cos ^ { 2 } x d x$

OCR FP2 2009 June Q9

14 marks Challenging +1.2

9

It is given that, for non-negative integers $n$, $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \mathrm {~d} \theta$$ Show that, for $n \geqslant 2$, $$n I _ { n } = ( n - 1 ) I _ { n - 2 } .$$
The equation of a curve, in polar coordinates, is $$r = \sin ^ { 3 } \theta , \quad \text { for } 0 \leqslant \theta \leqslant \pi$$
1. Find the equations of the tangents at the pole and sketch the curve.
2. Find the exact area of the region enclosed by the curve. RECOGNISING ACHIEVEMENT

OCR FP2 2013 June Q4

8 marks Challenging +1.2

4 It is given that $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \mathrm {~d} x$ for $n \geqslant 0$.

Show that $I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$ for $n \geqslant 2$.
Hence find $I _ { 11 }$ as an exact fraction.

CAIE FP1 2018 June Q9

10 marks Challenging +1.8

9

Using the substitution $u = \tan x$, or otherwise, find $\int \sec ^ { 2 } x \tan ^ { 2 } x \mathrm {~d} x$.
It is given that, for $n \geqslant 0$, $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \tan ^ { 2 } x \mathrm {~d} x$$
Using the result that $\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x ) = \tan x \sec x$, show that, for $n \geqslant 2$, $$( n + 1 ) I _ { n } = ( \sqrt { } 2 ) ^ { n - 2 } + ( n - 2 ) I _ { n - 2 }$$
Hence find the mean value of $\sec ^ { 4 } x \tan ^ { 2 } x$ with respect to $x$ over the interval $0 \leqslant x \leqslant \frac { 1 } { 4 } \pi$, giving your answer in exact form.

OCR Further Additional Pure 2021 November Q7

8 marks Challenging +1.2

7 Let $\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { \mathrm { n } } \mathrm { xdx }$ for integers $n \geqslant 0$.

Show that, for $n \geqslant 2 , \quad \mathrm { nl } _ { \mathrm { n } } = ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } - 2 }$.
Use this reduction formula to deduce the exact value of $I _ { 8 }$.
Use the results of parts (a) and (b) to determine the exact value of $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 6 } x \sin ^ { 2 } x d x$.

OCR Further Additional Pure Specimen Q5

9 marks Challenging +1.3

5 In this question you must show detailed reasoning.
It is given that $I _ { n } = \int _ { 0 } ^ { \pi } \sin ^ { n } \theta \mathrm {~d} \theta$ for $n \geq 0$.

Prove that $I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$ for $n \geq 2$.
(a) Evaluate $I _ { 1 }$.
(b) Use the reduction formula to determine the exact value of $\int _ { 0 } ^ { \pi } \cos ^ { 2 } \theta \sin ^ { 5 } \theta \mathrm {~d} \theta$.

Edexcel FP2 2019 June Q5

8 marks Challenging +1.8

5. $$I _ { n } = \int \operatorname { cosec } ^ { n } x \mathrm {~d} x \quad n \in \mathbb { Z }$$

Prove that, for $n \geqslant 2$ $$I _ { n } = \frac { n - 2 } { n - 1 } I _ { n - 2 } - \frac { \operatorname { cosec } ^ { n - 2 } x \cot x } { n - 1 }$$
Hence show that $$\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } \operatorname { cosec } ^ { 6 } x \mathrm {~d} x = \frac { 56 } { 135 } \sqrt { 3 }$$

Edexcel FP2 2022 June Q9

7 marks Challenging +1.8

9. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } 2 x d x$$

Prove that for $n \geqslant 2$ $$I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$$
Hence determine the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } 64 \sin ^ { 5 } x \cos ^ { 5 } x d x$$

Edexcel FP2 Specimen Q7

9 marks Challenging +1.2

7. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$

Prove that, for $n \geqslant 2$, $$n I _ { n } = ( n - 1 ) I _ { n - 2 }$$
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c262813-4160-4eda-9a36-e4ba38182c8a-22_588_1018_630_520} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A designer is asked to produce a poster to completely cover the curved surface area of a solid cylinder which has diameter 1 m and height 0.7 m . He uses a large sheet of paper with height 0.7 m and width of $\pi \mathrm { m }$.
Figure 2 shows the first stage of the design, where the poster is divided into two sections by a curve. The curve is given by the equation $$y = \sin ^ { 2 } ( 4 x ) - \sin ^ { 10 } ( 4 x )$$ relative to axes taken along the bottom and left hand edge of the paper.
The region of the poster below the curve is shaded and the region above the curve remains unshaded, as shown in Figure 2. Find the exact area of the poster which is shaded.

AQA Further Paper 2 Specimen Q8

5 marks Standard +0.8

Given that $I_n = \int_0^{\frac{\pi}{2}} \sin^n x \, dx$ \quad $n \geq 0$ show that $n I_n = (n-1)I_{n-2}$ \quad $n \geq 2$ [5 marks]

OCR Further Additional Pure 2017 Specimen Q5

9 marks Challenging +1.2

In this question you must show detailed reasoning. It is given that $I_n = \int_0^\pi \sin^n \theta \, d\theta$ for $n \geq 0$.

Prove that $I_n = \frac{n-1}{n} I_{n-2}$ for $n \geq 2$. [5]
Evaluate $I_1$ and use the reduction formula to determine the exact value of $\int_0^\pi \cos^2 \theta \sin^5 \theta \, d\theta$. [4]