Click on an arrow to get a description of the connection!
Click on an arrow to get a description of the connection!
Show requirements
| Concept |
Content |
| Derivatives |
Derivatives of elementary functions and rules for derivatives. |
| Integral |
Some properties of integrals |
Show consequences
As a reminder: Product rule of differentiation \((uv)' = u'v + uv'\)
Theorem 1. For \(u,v:
[a,b]\to\mathbb{R}\) continuously differentiable: \[\int u'(x) v(x)\,\mathrm{d} x = u(x) v(x) -
\int u(x)v'(x)\,\mathrm{d} x,\] and \[\int_{a}^b u'(x) v(x)\,\mathrm{d} x =
u(x)v(x)\bigr|_{x=a}^b - \int_a^b u(x)v'(x)\,\mathrm{d}
x.\]
Typical
applications
polynomial \(\cdot\) (\(\sin\), \(\cos\), \(\exp\), \(\sinh\), \(\cosh\))
\(v=\) polynomial, multiple application
possible
polynomial \(\cdot\) (\(\log\))
\(v=\log\), vanishes after
application
(\(\sin\), \(\cos\), \(\exp\), \(\sinh\), \(\cosh\)) \(\cdot\) (\(\sin\), \(\cos\), \(\exp\), \(\sinh\), \(\cosh\))
here also rearranging terms (trigonometric identities, Pythagoras)
required
Discuss your questions by typing below.