Reduction Formulae - Question Types

Trigonometric power reduction

A question is this type if and only if the integral I_n involves powers of sin(x), cos(x), tan(x), sec(x), cosec(x), or cot(x) and requires deriving or using a reduction formula specific to trigonometric functions.

17 Challenging +1.4

14.5% of questions

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8 Given that $I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x \quad n \geq 0$ show that $n I _ { n } = ( n - 1 ) I _ { n - 2 } \quad n \geq 2$ [0pt] [5 marks]

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Easiest question 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

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Hardest question 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.

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Algebraic function with square root

A question is this type if and only if I_n involves x^n multiplied or divided by a square root of a polynomial (e.g., √(a²-x²), √(x²+a²), √(ax-x²)).

11 Challenging +1.5

9.4% of questions

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4 It is given that $$I _ { n } = \int _ { 0 } ^ { 1 } \frac { x ^ { n } } { \sqrt { } ( 1 + 2 x ) } \mathrm { d } x$$ Show that, for $n \geqslant 1$, $$( 2 n + 1 ) I _ { n } = \sqrt { } 3 - n I _ { n - 1 }$$ Show that $$I _ { 3 } = \frac { 2 } { 35 } ( \sqrt { } 3 + 1 )$$

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Easiest question Challenging +1.2 »

5. Given that $$I _ { n } = \int x ^ { n } \sqrt { ( x + 8 ) } \mathrm { d } x , \quad n \geqslant 0 , x \geqslant 0$$

show that, for $n \geqslant 1$ $$I _ { n } = \frac { p x ^ { n } ( x + 8 ) ^ { \frac { 3 } { 2 } } } { 2 n + 3 } - \frac { q n } { 2 n + 3 } I _ { n - 1 }$$ where $p$ and $q$ are constants to be found.
Use part (a) to find the exact value of $$\int _ { 0 } ^ { 10 } x ^ { 2 } \sqrt { ( x + 8 ) } d x$$ giving your answer in the form $k \sqrt { 2 }$, where $k$ is rational.

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Hardest question Challenging +1.8 »

8. $$I _ { n } = \int \frac { x ^ { n } } { \sqrt { \left( x ^ { 2 } + k ^ { 2 } \right) } } \mathrm { d } x \quad \text { where } k \text { is a constant and } n \in \mathbb { Z } ^ { + }$$

Show that, for $n \geqslant 2$ $$I _ { n } = \frac { x ^ { n - 1 } } { n } \left( x ^ { 2 } + k ^ { 2 } \right) ^ { \frac { 1 } { 2 } } - \frac { ( n - 1 ) } { n } k ^ { 2 } I _ { n - 2 }$$
Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { \sqrt { \left( x ^ { 2 } + 1 \right) } } \mathrm { d } x$$

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Polynomial times trigonometric

A question is this type if and only if I_n involves x^n or a polynomial multiplied by sin(x), cos(x), sin(kx), cos(kx), or similar trigonometric functions (not powers of trig functions).

9 Challenging +1.3

7.7% of questions

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6 It is given that, for non-negative integers $n$, $$I _ { n } = \int _ { 0 } ^ { \pi } x ^ { n } \sin x \mathrm {~d} x$$

Prove that, for $n \geqslant 2 , I _ { n } = \pi ^ { n } - n ( n - 1 ) I _ { n - 2 }$.
Find $I _ { 5 }$ in terms of $\pi$.

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Easiest question Challenging +1.2 »

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

Prove that, for $n \geqslant 2$, $$I _ { n } = \frac { 1 } { 4 } n \left( \frac { \pi } { 4 } \right) ^ { n - 1 } - \frac { 1 } { 4 } n ( n - 1 ) I _ { n - 2 }$$
Find the exact value of $I _ { 2 }$
Show that $I _ { 4 } = \frac { 1 } { 64 } \left( \pi ^ { 3 } - 24 \pi + 48 \right)$

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Hardest question Challenging +1.8 »

6. $$I _ { n } = \int _ { 0 } ^ { \sqrt { \frac { \pi } { 2 } } } x ^ { n } \cos \left( x ^ { 2 } \right) \mathrm { d } x \quad n \geqslant 1$$

Prove that, for $n \geqslant 5$ $$I _ { n } = \frac { 1 } { 2 } \left( \frac { \pi } { 2 } \right) ^ { \frac { n - 1 } { 2 } } - \frac { 1 } { 4 } ( n - 1 ) ( n - 3 ) I _ { n - 4 }$$
Hence, determine the exact value of $I _ { 5 }$, giving your answer in its simplest form.

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Compound expressions with binomial expansion

A question is this type if and only if the integrand involves (a+bx)^n or similar binomial expressions that may require expansion or special reduction techniques.

9 Challenging +1.4

7.7% of questions

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6

Given that $I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \sqrt { } ( 1 - x ) \mathrm { d } x$, prove that, for $n \geqslant 1$, $$( 2 n + 3 ) I _ { n } = 2 n I _ { n - 1 } .$$
Hence find the exact value of $I _ { 2 }$.

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Easiest question Challenging +1.2 »

4. $$I _ { n } = \int _ { 0 } ^ { \sqrt { 3 } } \left( 3 - x ^ { 2 } \right) ^ { n } \mathrm {~d} x , \quad n \geqslant 0$$

Show that, for $n \geqslant 1$ $$I _ { n } = \frac { 6 n } { 2 n + 1 } I _ { n - 1 }$$
Hence find the exact value of $I _ { 4 }$, giving your answer in the form $k \sqrt { 3 }$ where $k$ is a rational number to be found.

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Hardest question Challenging +1.8 »

8. $$I _ { n } = \int _ { 0 } ^ { k } x ^ { n } ( k - x ) ^ { \frac { 1 } { 2 } } \mathrm {~d} x \quad n \geqslant 0$$ where $k$ is a positive constant.

Show that $$I _ { n } = \frac { 2 k n } { 3 + 2 n } I _ { n - 1 } \quad n \geqslant 1$$ Given that $$\int _ { 0 } ^ { k } x ^ { 2 } ( k - x ) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = \frac { 9 \sqrt { 3 } } { 280 }$$
use the result in part (a) to determine the exact value of $k$.

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Method of differences

A question is this type if and only if it derives a formula of the form I_n - I_(n-k) = f(n) and uses it to find sums or specific values.

9 Challenging +1.6

7.7% of questions

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

By considering $I _ { n } + I _ { n - 2 }$, or otherwise, show that, for $n \geqslant 2$, $$( n - 1 ) \left( I _ { n } + I _ { n - 2 } \right) = 1 .$$
Find $I _ { 4 }$ in terms of $\pi$.

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Easiest question Standard +0.8 »

5 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { n } x \mathrm {~d} x$$ where $n \geqslant 0$. Use the fact that $\tan ^ { 2 } x = \sec ^ { 2 } x - 1$ to show that, for $n \geqslant 2$, $$I _ { n } = \frac { 1 } { n - 1 } - I _ { n - 2 }$$ Show that $I _ { 8 } = \frac { 1 } { 7 } - \frac { 1 } { 5 } + \frac { 1 } { 3 } - 1 + \frac { 1 } { 4 } \pi$.

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Hardest question Challenging +1.8 »

8. $$I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { 2 n } x \mathrm {~d} x , \quad n \geqslant 0$$

Show that, for $n \geqslant 1$
Hence show that $$\int _ { 0 } ^ { \ln 2 } \tanh ^ { 4 } x \mathrm {~d} x = p + \ln 2$$ where $p$ is a rational number to be found.
8. $\quad I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { 2 n } x \mathrm {~d} x , \quad n \geqslant 0$
Show that, for $n \geqslant 1$ $$I _ { n } = I _ { n - 1 } - \frac { 1 } { 2 n - 1 } \left( \frac { 3 } { 5 } \right) ^ { 2 n - 1 }$$

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Prove formula by induction

A question is this type if and only if it asks to prove a formula for I_n using mathematical induction after establishing a reduction formula.

8 Challenging +1.4

6.8% of questions

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7 Let $$I _ { n } = \int _ { 0 } ^ { 1 } t ^ { n } \mathrm { e } ^ { - t } \mathrm {~d} t$$ where $n \geqslant 0$. Show that, for all $n \geqslant 1$, $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 }$$ Hence prove by induction that, for all positive integers $n$, $$I _ { n } < n ! .$$

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Rational function powers

A question is this type if and only if I_n involves (1+x²)^(-n), (a+x²)^(-n), or similar rational function powers.

8 Challenging +1.7

6.8% of questions

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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 $I _ { 2 }$

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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 Challenging +1.3

6.0% of questions

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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.

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Derive reduction formula by differentiation

A question is this type if and only if it asks to derive a reduction formula by considering the derivative of a given product expression (e.g., d/dx of x(1-x²)^n).

7 Challenging +1.7

6.0% of questions

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4 The integral $\mathrm { I } _ { \mathrm { n } }$ is defined by $\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 1 } \left( 1 + \mathrm { x } ^ { 5 } \right) ^ { \mathrm { n } } \mathrm { dx }$.

By considering $\frac { d } { d x } \left( x \left( 1 + x ^ { 5 } \right) ^ { n } \right)$, or otherwise, show that $$( 5 n + 1 ) l _ { n } = 2 ^ { n } + 5 n l _ { n - 1 }$$
Find the exact value of $I _ { 3 }$.

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Hyperbolic function reduction

A question is this type if and only if the integral I_n involves hyperbolic functions (sinh, cosh, tanh, sech) and requires deriving or using a reduction formula.

5 Challenging +1.8

4.3% of questions

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

Show that, for $n \geqslant 2$ $$n I _ { n } = \sinh x \cosh ^ { n - 1 } x + ( n - 1 ) I _ { n - 2 }$$
Hence find the exact value of $$\int _ { 0 } ^ { \ln 2 } \cosh ^ { 5 } x \mathrm {~d} x$$

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Exponential times polynomial

A question is this type if and only if I_n involves e^(ax) or e^(f(x)) multiplied by x^n or a polynomial in x, where the exponential is the primary function being integrated with a polynomial factor.

4 Challenging +1.0

3.4% of questions

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4 You are given that $I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { 2 x } \mathrm {~d} x$ for $n \geqslant 0$.

Show that $I _ { n } = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } - \frac { 1 } { 2 } n I _ { n - 1 }$ for $n \geqslant 1$.
Find $I _ { 3 }$ in terms of e.

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Logarithmic power integrals

A question is this type if and only if I_n involves (ln x)^n or x^n(ln x)^m as the integrand.

4 Challenging +1.4

3.4% of questions

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5 It is given that, for non-negative integers $n$, $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x$$

Show that, for $n \geqslant 1$, $$I _ { n } = \mathrm { e } - n I _ { n - 1 } .$$
Find $I _ { 3 }$ in terms of e.

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Mixed trigonometric products

A question is this type if and only if I_n involves products of different trigonometric functions with different powers (e.g., sin^m(x)cos^n(x)) requiring a two-index reduction formula.

3 Challenging +1.4

2.6% of questions

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5 Let $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \sin ^ { 2 } x \mathrm {~d} x$, for $n \geqslant 0$. By differentiating $\cos ^ { n - 1 } x \sin ^ { 3 } x$ with respect to $x$, prove that $$( n + 2 ) I _ { n } = ( n - 1 ) I _ { n - 2 } \quad \text { for } n \geqslant 2$$ Hence find the exact value of $I _ { 4 }$.

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Establish bounds or inequalities

A question is this type if and only if it uses reduction formula results to prove inequalities or establish bounds on I_n or related constants (e.g., bounds on π or e).

3 Challenging +1.3

2.6% of questions

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7 It is given that, for non-negative integers $n , I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } x \mathrm {~d} x$.

Show that $I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$ for $n \geqslant 2$.
Explain why $I _ { 2 n + 1 } < I _ { 2 n - 1 }$.
It is given that $I _ { 2 n + 1 } < I _ { 2 n } < I _ { 2 n - 1 }$. Take $n = 5$ to find an interval within which the value of $\pi$ lies.

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Centroid and mean value

A question is this type if and only if it applies a reduction formula to find centroid coordinates, mean values, or center of mass.

2 Challenging +1.2

1.7% of questions

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9 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \mathrm {~d} \theta$$ where $n$ is a non-negative integer. Show that $I _ { n + 2 } = \frac { n + 1 } { n + 2 } I _ { n }$. The region $R$ of the $x - y$ plane is bounded by the $x$-axis, the line $x = \frac { \pi } { 2 m }$ and the curve whose equation is $y = \sin ^ { 4 } m x$, where $m > 0$. Find the $y$-coordinate of the centroid of $R$.

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Exponential times trigonometric power

A question is this type if and only if I_n involves e^(ax) multiplied by sin^n(x), cos^n(x), or other powers of trigonometric functions.

2 Challenging +1.8

1.7% of questions

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6. $$I _ { n } = \int \mathrm { e } ^ { x } \sin ^ { n } x \mathrm {~d} x \quad n \in \mathbb { Z } \quad n \geqslant 0$$

Show that $$I _ { n } = \frac { \mathrm { e } ^ { x } \sin ^ { n - 1 } x } { n ^ { 2 } + 1 } ( \sin x - n \cos x ) + \frac { n ( n - 1 ) } { n ^ { 2 } + 1 } I _ { n - 2 } \quad n \geqslant 2$$
Hence find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } e ^ { x } \sin ^ { 4 } x d x$$ giving your answer in the form $A \mathrm { e } ^ { \frac { \pi } { 2 } } + B$ where $A$ and $B$ are rational numbers to be determined.

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Volume and surface area of revolution

A question is this type if and only if it applies a reduction formula to calculate volume of revolution, surface area of revolution, or arc length of a curve.

1 Challenging +1.8

0.9% of questions

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Find the value of $I _ { 2 }$.
Show that, for $n > 2$, $$( n - 1 ) I _ { n } = 2 ^ { \frac { 1 } { 2 } n - 1 } + ( n - 2 ) I _ { n - 2 }$$
The curve $C$ has equation $y = \sec ^ { 3 } x$ for $0 \leqslant x \leqslant \frac { 1 } { 4 } \pi$. The region $R$ is bounded by $C$, the $x$-axis, the $y$-axis and the line $x = \frac { 1 } { 4 } \pi$. Find the volume of revolution generated when $R$ is rotated through $2 \pi$ radians about the $x$-axis.

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Parametric or substitution setup

A question is this type if and only if it requires using a substitution (given or to be found) to transform the integral into a form involving I_n or to evaluate it.

1 Challenging +1.8

0.9% of questions

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Let $I _ { n } = \int _ { 1 } ^ { \sqrt { } 2 } \left( x ^ { 2 } - 1 \right) ^ { n } \mathrm {~d} x$.

Show that, for $n \geqslant 1$, $$( 2 n + 1 ) I _ { n } = \sqrt { } 2 - 2 n I _ { n - 1 } .$$
Using the substitution $x = \sec \theta$, show that $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { 2 n + 1 } \theta \sec \theta \mathrm {~d} \theta$$
Deduce the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 7 } \theta } { \cos ^ { 8 } \theta } \mathrm {~d} \theta$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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Moment of inertia

A question is this type if and only if it applies integration (often with reduction formula techniques) to calculate moment of inertia of rigid bodies about various axes.

0

0.0% of questions

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6. Three equal uniform rods, each of mass $m$ and length $2 a$, form the sides of a rigid equilateral triangular frame $A B C$. The frame is free to rotate in a vertical plane about a fixed smooth horizontal axis $L$ which passes through $A$ and is perpendicular to the plane of the frame.

Show that the moment of inertia of the frame about $L$ is $6 m a ^ { 2 }$. The frame is held with $A B$ horizontal and $C$ below $A B$, and released from rest. Given that the centre of mass of the frame is two thirds of the way along a median from a vertex,
find the magnitude of the force exerted by the axis on the frame at $A$ at the instant when the frame is released.

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Definite integral with special limits

A question is this type if and only if the reduction formula involves definite integrals with limits that produce specific numerical values (like powers of 2, 3, or expressions involving √2, √3) in the recurrence relation.

0

0.0% of questions

Apply reduction formula iteratively

A question is this type if and only if it asks to use a given or derived reduction formula repeatedly to find I_n for a larger value of n (typically n≥4), showing each step.

0

0.0% of questions

Basic integral evaluation

A question is this type if and only if it asks to evaluate I_n for a specific small value of n (typically n=1, 2, or 3) without deriving a reduction formula.

0

0.0% of questions

Unclassified

Questions not yet assigned to a type.

7

6.0% of questions

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10 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \mathrm {~d} x$$ where $n \geqslant 0$. Show that, for all $n \geqslant 2$, $$I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$$ A curve has parametric equations $x = a \sin ^ { 3 } t$ and $y = a \cos ^ { 3 } t$, where $a$ is a constant and $0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$. Show that the mean value $m$ of $y$ over the interval $0 \leqslant x \leqslant a$ is given by $$m = 3 a \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( \cos ^ { 4 } t - \cos ^ { 6 } t \right) \mathrm { d } t$$ Find the exact value of $m$, in terms of $a$.

4 Let $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { n } } \mathrm {~d} x$. Prove that, for every positive integer $n$, $$2 n I _ { n + 1 } = 2 ^ { - n } + ( 2 n - 1 ) I _ { n }$$ Given that $I _ { 1 } = \frac { 1 } { 4 } \pi$, find the exact value of $I _ { 3 }$.

10 It is given that $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 2 n } x } { \cos x } \mathrm {~d} x$, where $n \geqslant 0$. Show that $$I _ { n } - I _ { n + 1 } = \frac { 2 ^ { - \left( n + \frac { 1 } { 2 } \right) } } { 2 n + 1 }$$ Hence show that $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 6 } x } { \cos x } \mathrm {~d} x = \ln ( 1 + \sqrt { } 2 ) - \frac { 73 } { 120 } \sqrt { } 2$.

7 Let $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x$, where $n$ is a non-negative integer. Show that $$I _ { n } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 } , \quad \text { for } n \geqslant 2$$ Find the exact value of $I _ { 4 }$.

6 Let $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x$.

Prove that, for $n \geqslant 2$, $$I _ { n } + n ( n - 1 ) I _ { n - 2 } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } .$$
Calculate the exact value of $I _ { 1 }$ and deduce the exact value of $I _ { 3 }$.

7 Let $I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x$. Show that, for all positive integers $n$, $$I _ { n } = n I _ { n - 1 } - 1$$ Find the exact value of $I _ { 4 }$. By considering the area of the region enclosed by the $x$-axis, the $y$-axis and the curve with equation $y = ( 1 - x ) ^ { 4 } \mathrm { e } ^ { x }$ in the interval $0 \leqslant x \leqslant 1$, show that $$\frac { 65 } { 24 } < \mathrm { e } < \frac { 11 } { 4 }$$

9 It is given that $I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x$ for $n \geqslant 0$. Show that $$I _ { n } = ( n - 1 ) \left[ I _ { n - 2 } - I _ { n - 1 } \right] \text { for } n \geqslant 2$$ Hence find, in an exact form, the mean value of $( \ln x ) ^ { 3 }$ with respect to $x$ over the interval $1 \leqslant x \leqslant \mathrm { e }$.