Search for notes by fellow students, in your own course and all over the country.
Browse our notes for titles which look like what you need, you can preview any of the notes via a sample of the contents. After you're happy these are the notes you're after simply pop them into your shopping cart.
My Basket
(Oxford) Solutions for B5: General Relativity and Cosmology, 2011-2016£6.24
Or: Edit My Basket
Title: Integration by parts and by magic substitution
Description: Easy way to learn Integration by parts and by magic substitution with proof and examples.
Description: Easy way to learn Integration by parts and by magic substitution with proof and examples.
Document Preview
Extracts from the notes are below, to see the PDF you'll receive please use the links above
Integration by Parts: Method
From
d(fg) = fdg + gdf
upon integrating both sides and rearranging, we get
Use of this statement is called "integration by parts"
...
These occur when the integrand is a monomial times an exponent or
trigonometric function, or when the integrand has logarithms in it
...
It also helps for inverse powers of sine and cosine
...
Here and in the other examples, we choose C = 0
...
Use
to rewrite
Obtain
Rearrange
Iterate this to obtain
When n is even this stops automatically; when n is odd, the result is in our table
of simple trigonometric integrals for n = 1:
The Magic Substitution z = tan(x/2)
Suppose our integrand is a rational function of sin(x) and cos(x)
...
Theorem:
If z = tan(x / 2), then
,
,
and
and any rational function of xdx becomes a rational function of zdz
Title: Integration by parts and by magic substitution
Description: Easy way to learn Integration by parts and by magic substitution with proof and examples.
Description: Easy way to learn Integration by parts and by magic substitution with proof and examples.