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Advanced Integration Techniques

The document outlines key topics in integration that will be covered, including: 1) integration techniques from A-level math syllabi, 2) differentiation under the integral sign, 3) the Weierstrass substitution, and 4) using infinite series to evaluate integrals. It also covers 5) the reflection property of integrals, 6) the floor function, 7) properties of the gamma function, 8) the Riemann zeta function, and 9) even and odd functions.

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55 views1 page

Advanced Integration Techniques

The document outlines key topics in integration that will be covered, including: 1) integration techniques from A-level math syllabi, 2) differentiation under the integral sign, 3) the Weierstrass substitution, and 4) using infinite series to evaluate integrals. It also covers 5) the reflection property of integrals, 6) the floor function, 7) properties of the gamma function, 8) the Riemann zeta function, and 9) even and odd functions.

Uploaded by

Obama binladen
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Integration Competition Syllabus

• Everything which is on the STEP/A level Maths and Further Maths syl-
labus for Integration.
• Differentiation under the integral sign (DUTIS):
Z b ! Z
b
d ∂
f (x, t)dx = (f (x, t)) dx
dt a a ∂t

x
• The Weierstrass substitution, t = tan (also known as t substitution)
2
• Infinite series and their use in evaluating integrals, swapping an integral
and an infinite sum. Convergence issues won’t be considered.
• The reflection property of integrals:
Z b Z b
f (x)dx = f (a + b − x)dx
a a

• The floor function bxc which rounds down to the integer less than or equal
to x.
Z ∞
• The gamma function Γ(n) = xn−1 e−x dx; knowledge of the properties
0
Γ(1) = 1, Γ(n) = (n − 1)! - that the gamma function is an extension of
the factorials to non integer arguments.
∞
X 1
• The Riemann zeta function ζ(s) = for s > 1.
n=1
ns

• Odd functions, functions such that f (−x) = −f (x) and even functions,
functions such that f (−x) = f (x).

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