Integration

9 methods · 33 questions

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A term defined by an integral that depends on an index is usually constant or follows a simple closed form. Evaluate the integral as a function of the index (or compute the first two or three terms to spot the pattern), substitute that closed form back, and turn the remaining sum into a standard arithmetic, geometric or triangular series.

Trigger: a sequence $u_{n}$ or function $f_{n}$ defined by an integral, or $r \sum \int \dots$ with the index inside the integrand or limits.

Instances:
(i) compute $u_{1}, u_{2}, u_{3}$ from the recurrence and recognise a power or factorial pattern, then confirm by induction;
(ii) evaluate the inner integral to a formula in $n$ (often a constant, $n /2$ , or $r (r + 1) /2$ ) before summing;
(iii) split an integral over $[0, r]$ into unit blocks whose values are given, reducing it to summing those block values.

Linked questions (3)

TMUA 2017 Paper 1 · Q17

The two functions

F (n)

and

G (n)

are defined as follows for positive integers

n

F (n) = \frac{1}{n} \int_{0}^{n} (n - x) d x, G (n) = r = 1 \sum n F (r) .

What is the smallest positive integer

n

such that

G (n) > 150

A $22$
B $23$
C $24$
D $25$
E $26$

TMUA 2017 Paper 2 · Q6

A sequence

u_{0}, u_{1}, u_{2}, \dots

is defined as follows:

u_{0} = 1, u_{n} = \int_{0}^{1} 4 x u_{n - 1} d x for n ⩾ 1.

What is the value of

u_{1000}

A $2^{1000}$
B $4^{1000}$
C $\frac{4}{1000 !}$
D $\frac{4}{1001 !}$
E $\frac{2 ^{1000}}{1000 !}$
F $\frac{4 ^{1000}}{1000 !}$
G $\frac{2 ^{1000}}{1001 !}$
H $\frac{4 ^{1000}}{1001 !}$

TMUA 2021 Paper 1 · Q13

The function f is such that, for every integer

n

\int_{n}^{n + 1} f (x) d x = n + 1.

Evaluate

r = 1 \sum 8 (\int_{0}^{r} f (x) d x)

A $36$
B $84$
C $120$
D $165$
E $204$
F $288$